#6660661

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660402
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,32)(27,31,34,38)(28,30,36,33)(35,37,40,39);(3,7)(5,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(27,34)(31,38)(35,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,30)(28,36)(31,37)(32,33)(35,40)(38,39);(3,7)(5,8)(10,16)(12,20)(14,17)(18,24)(25,27)(26,31)(28,35)(29,34)(30,37)(32,38)(33,39)(36,40);(2,6)(5,8)(10,16)(14,17)(15,22)(21,23)(26,32)(27,34)(30,33)(35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567857);