#124176

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1459
prime radical	#392	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 393 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31372);