#124199

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1459
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5,6,7)(9,10,13,19,14,18,27,33,15,20,26,34,12,16,25,29)(11,17,22,31,23,30,37,39,24,32,36,40,21,28,35,38);(1,2)(3,7)(4,6)(5,8)(10,16,20,18)(13,26)(17,28,32,30)(19,29,34,33)(22,36)(25,27)(31,38,40,39)(35,37);(9,11,14,23,15,24,12,21)(10,17,18,30,20,32,16,28)(13,22,27,37,26,36,25,35)(19,31,33,39,34,40,29,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gi,e^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,d^b=dg,e^b=efgi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31395);