#6657573

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632068
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
prime radical	#35521	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6564769);