#7092005

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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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)(23,28)(24,31)(27,35)(30,43)(32,40)(33,41)(34,47)(36,39)(38,44)(42,45)(46,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,31,23,21,35,36)(18,24,27,29,28,39,37,26)(20,30,34,44,33,32,45,46)(25,38,42,43,41,48,47,40);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,33)(18,25,28,41)(19,30,21,32)(22,34,35,45)(24,38,39,48)(26,40,29,43)(27,42,37,47)(31,44,36,46);(11,16)(13,15)(17,21,23,19)(18,26,28,29)(20,32,33,30)(22,31,35,36)(24,27,39,37)(25,40,41,43)(34,44,45,46)(38,42,48,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,33)(25,41)(30,32)(34,45)(38,48)(40,43)(42,47)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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=dfi,e^b=eh,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, 6999201);