GROUP OF ORDER 32 # 46

# GROUP # 46

The MAGMA library number for G is 6

### GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.3, G.2^G.1 = G.2 * G.4, G.3^G.2 = G.3 * G.5, G.4^G.1 = G.4 * G.5

The center of G is abelian of type [ 2 ]
The orders of the terms of the lower central series are [ 32, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 2, 8, 32 ]
The order of the Frattini subgroup is 8
The exponent of G is 4
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 8, 8 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 32, 32 ]

### The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u, t, s ] in degrees [ 1, 1, 2, 2, 2, 3, 3, 4 ] by the ideal generated by the relations [ z^2, z*y, y*x + z*v, z*w + z*v, y*w + z*v, w^2 + x*v, z*u, y*u, z*t, x*u + v*u + x*t + w*t, w*u + v*u + w*t, x^2*v + v^3 + y*v*t + y^2*s + t^2, x*w*v + w*v^2 + u^2 + u*t, x*v^2 + v^3 + y*v*t + y^2*s + u^2 + t^2 ]

The Hilbert series for the cohomology ring is -1/(t^4 - 2*t^3 + 2*t - 1)
Its denominator factors as ( t - 1 )^3 ( t + 1 )^1

The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [ s, y^2 + x ]
A homogeneous set of parameters is the set [ s, y^2 + x, v ] of degrees [ 4, 2, 2 ]

### The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ y, z ] [ x, w ] [ u, t ] [ y*t ] [ w*t ]
ROW (1) [] [] [ z*v ]
The spectral sequence satisfies Poincaré duality

# Restrictions to Maximal Subgroups

## The essential cohomology of G is zero

### CHECKS

paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true