#6810051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6717247);