#6811507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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)(3,6)(4,8)(5,7)(9,10)(11,16)(12,14)(13,27)(15,19)(17,31)(18,29)(20,24)(21,36)(22,25)(23,33)(26,40)(28,39)(30,35)(32,38)(34,37);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,25)(13,28)(14,22)(15,24)(17,34)(18,33)(19,20)(21,30)(23,29)(26,32)(27,39)(31,37)(35,36)(38,40);(9,12,14,19,15,18,29,10)(11,20,23,25,24,16,22,33)(13,26,30,27,31,35,40,17)(21,36,28,39,38,32,37,34);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,13,29,40,15,31,14,30)(10,17,18,35,19,27,12,26)(11,21,22,37,24,38,23,28)(16,32,25,39,20,36,33,34)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=f,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,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, 6718703);