#125929

order	512
nilpotency class	4
isoclinism	isoclinic class of #125725
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	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

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

normal subgroups only

This group has 339 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)(11,15)(12,16)(13,28)(14,18)(17,27)(19,26)(20,29)(21,38)(22,31)(23,39)(24,34)(25,33)(30,37)(32,36)(35,40);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,31)(12,23,25,20)(13,21,26,36)(16,29,33,39)(17,30,34,40)(19,32,28,38)(24,37,27,35);(10,16,18,33)(12,24)(13,26)(15,29,31,39)(17,28,34,19)(20,35)(21,36)(23,37)(25,27)(30,38,40,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,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=ehi,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, 33125);