#6667456

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1205 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,e^b=ei,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, 6574652);