#6643306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632008
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550502);