#6649101

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636816
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26)(27,34)(28,32)(29,38)(30,33)(31,35)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23);(3,8)(5,7)(25,27,28,36,30,37,29,35)(26,31,38,39,33,40,32,34);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,40,34);(10,16)(14,17)(15,20)(23,24)(25,28,30,29)(26,32,33,38)(27,36,37,35)(31,34,40,39)

Polycyclic presentation:

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

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, 6556297);