#6485534

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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)(17,18,22,25)(19,27,29,43)(20,31,34,44)(21,35,38,26)(23,36,40,30)(24,37,42,33)(28,39,32,41)(45,47,48,46);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,32)(21,36)(22,29)(24,41)(25,40)(26,43)(27,35)(28,34)(30,38)(31,48)(33,46)(37,47)(39,42)(44,45);(10,14)(15,16)(17,20,22,34)(18,24,25,42)(19,28,29,32)(21,37,38,33)(23,39,40,41)(26,44,35,31)(27,45,43,48)(30,47,36,46);(9,11,13,12)(10,15,14,16)(19,29)(23,40)(27,43)(28,32)(30,36)(39,41)(45,48)(46,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,25)(19,29)(24,42)(26,35)(28,32)(30,36)(31,44)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392730);