#6667429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660376
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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)(25,26,30,34,29,31,28,33)(27,35,37,39,36,40,38,32);(2,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,24,23)(26,31)(27,36)(33,34)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,17,24)(11,19,13,21)(15,22,18,23)(25,27,29,36)(26,32,31,39)(28,37,30,38)(33,35,34,40);(2,5)(6,8)(10,17)(12,20)(15,18)(19,21)(25,28,29,30)(26,33,31,34)(27,37,36,38)(32,35,39,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,18,17,15)(12,21,20,19)(16,23,24,22)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,d^c=dfh,g^c=gi

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, 6574625);