#125692

order	512
nilpotency class	4
isoclinism	isoclinic class of #125644
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#10032	C₂
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=egi,d^c=dg,e^c=eh,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32888);