#7083207

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	#42685	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(17,18,22,26)(19,27,31,43)(20,24,33,40)(21,35,28,25)(23,36,38,30)(29,42,32,47)(34,48,44,41)(37,45,39,46);(9,10)(11,15)(12,14)(13,16)(17,19,28,36,22,31,21,30)(18,23,35,43,26,38,25,27)(20,32,44,46,33,29,34,45)(24,39,48,42,40,37,41,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,22,33)(18,24,26,40)(19,29,31,32)(21,34,28,44)(23,37,38,39)(25,41,35,48)(27,42,43,47)(30,46,36,45);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(19,30)(20,33)(21,28)(23,27)(24,40)(25,35)(29,45)(31,36)(32,46)(37,47)(38,43)(39,42);(10,14)(13,15)(19,31)(20,33)(23,38)(24,40)(27,43)(30,36)(34,44)(41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gh,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 6990403);