#124511

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10,13,19,14,17,27,34,15,20,28,37,12,18,25,35)(11,21,23,38,22,16,30,31,24,32,39,36,29,33,26,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,26)(14,29)(15,24)(17,21)(18,32)(19,36)(20,33)(23,25)(27,39)(28,30)(31,34)(35,40)(37,38);(9,12,15,14)(10,17,20,18)(11,22,24,29)(13,27,28,25)(16,21,33,32)(19,35,37,34)(23,26,39,30)(31,36,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31707);