#125885

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 56 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)(11,19)(12,23)(13,18)(14,28)(15,21)(16,27)(17,26)(20,35)(22,39)(24,34)(25,36)(29,33)(30,40)(31,37)(32,38);(2,5)(3,8)(6,7)(16,32)(19,35)(23,36)(33,39);(9,11,13,20)(10,15,18,31)(12,24,25,40)(14,29,26,22)(16,33,32,39)(17,34,28,30)(19,36,35,23)(21,38,37,27)

Polycyclic presentation:

2^9|a^2=f,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=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, 33081);