#7091885

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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

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

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, 6999081);