#6485694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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)(19,29)(20,34)(21,26)(22,37)(23,28)(24,32)(25,35)(27,39)(30,40)(31,46)(33,44)(36,43)(38,42)(41,48)(45,47);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,33)(18,24,28,42)(20,30,21,31)(22,38,39,32)(25,40,43,46)(26,41,34,45)(27,44,37,29)(35,48,36,47);(10,14)(15,16)(17,20,23,21)(18,25,28,43)(19,30,33,31)(22,36,39,35)(24,40,42,46)(26,37,34,27)(29,45,44,41)(32,48,38,47);(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,28,34)(19,31,33,30)(22,36,39,35)(24,41,42,45)(25,37,43,27)(29,46,44,40)(32,48,38,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(20,21)(25,43)(26,34)(30,31)(35,36)(40,46)(41,45)(47,48)

Polycyclic presentation:

2^9|b^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,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei

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, 6392890);