#6643316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632036
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(25,26,29,33,28,32,37,31)(27,34,36,39,35,30,38,40);(1,2,4,5)(3,6,7,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,28,35)(26,30,32,34)(29,36,37,38)(31,39,33,40);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,31)(29,37)(30,39)(32,33)(34,40)(36,38);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,35)(30,34)(36,38)(39,40);(10,17)(15,18)(16,21)(23,24)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550512);