#6755945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282613, #72282830, #72283138
maximal normal subgroup of	#72282613, #72282830, #72283138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6663141);