#126047

order	512
nilpotency class	4
isoclinism	isoclinic class of #125709
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,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=dfi,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, 33243);