#6808759

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

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,25)(20,34)(21,27)(22,39)(23,28)(24,29)(26,35)(30,46)(31,42)(32,44)(33,45)(36,47)(37,48)(38,41)(40,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,21)(22,32)(23,40)(24,33)(26,46)(27,34)(28,43)(29,45)(30,35)(31,37)(36,38)(39,44)(41,47)(42,48);(17,20,23,35,24,36,22,31)(18,26,28,47,29,42,39,34)(19,30,32,21,33,37,40,38)(25,41,44,46,45,27,43,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15);(3,11)(6,9)(10,16)(14,15)(17,21,23,37,24,38,22,30)(18,27,28,48,29,41,39,46)(19,31,32,20,33,35,40,36)(25,42,44,34,45,26,43,47)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=f,e^2=h,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, 6715955);