#7092046

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, 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

1326 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)(18,24)(21,22)(25,39)(26,32)(27,37)(28,41)(30,38)(34,35)(40,48)(42,45)(43,44)(46,47);(9,10)(11,15)(12,14)(13,16)(17,18,21,37,23,27,22,24)(19,28,38,39,31,25,30,41)(20,32,34,44,36,43,35,26)(29,45,47,40,33,48,46,42);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,33)(21,38)(22,30)(23,31)(24,39)(26,42)(27,28)(29,36)(32,45)(34,46)(35,47)(37,41)(40,44)(43,48);(11,16)(13,15)(17,20,23,36)(18,26,27,44)(19,29,31,33)(21,35,22,34)(24,32,37,43)(25,40,28,42)(30,47,38,46)(39,48,41,45);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,24,27,37)(19,30,31,38)(20,34,36,35)(25,41,28,39)(26,43,44,32)(29,46,33,47)(40,48,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=ghi,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=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=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, 6999242);