#6994552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6901748);