#6570256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,12,13,16)(17,18)(19,28)(20,32)(21,26)(22,27)(23,38)(24,30)(25,36)(29,44)(31,41)(33,47)(34,43)(35,46)(37,40)(39,48)(42,45);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,12)(17,19,22,30)(18,24,27,28)(20,33,35,29)(21,37,23,31)(25,42,43,39)(26,41,38,40)(32,44,46,47)(34,45,36,48);(9,12,15,16)(10,11,14,13)(17,20,21,34,22,35,23,36)(18,25,38,46,27,43,26,32)(19,29,37,48,30,33,31,45)(24,39,40,44,28,42,41,47);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,16,14,12)(19,30)(24,28)(29,33)(31,37)(39,42)(40,41)(44,47)(45,48);(1,4,5,3)(2,6,8,7)(9,14,15,10)(11,12,13,16)(17,21,22,23)(18,26,27,38)(19,31,30,37)(20,34,35,36)(24,40,28,41)(25,32,43,46)(29,45,33,48)(39,44,42,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477452);