#6477893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6385089);