#93951

order	512
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₈
Frattini, cosocle	C₄⋊#95	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₄⋊#95 ◁ G
LECS	G ▷ C₄⋊#95 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

(2,5,6,7)(9,10,11,13,12,14,15,16)(17,18)(19,26,27,22)(20,24,29,23)(21,25)(28,31,32,30);(1,2)(3,7)(4,6)(5,8)(17,19,20,27,21,28,29,32)(18,22,24,26,25,30,23,31)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=defhi,f^b=fi,h^b=hi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1147);