#6808120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=fi,e^2=hi,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

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, 6715316);