#7091226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6998422);