#7083402

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,35)(21,27)(22,28)(23,40)(24,29)(25,33)(26,31)(32,44)(34,48)(36,43)(37,46)(38,47)(39,45)(41,42);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33,24,34,22,32)(18,25,28,30,29,44,40,48)(20,21,36,37,31,38,39,41)(26,42,43,47,35,46,45,27);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,31)(18,26,29,35)(19,21,34,38)(22,39,23,36)(25,42,44,46)(27,30,47,48)(28,43,40,45)(32,41,33,37);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,31)(20,34)(22,37)(23,41)(24,38)(25,43)(26,30)(28,46)(29,47)(32,39)(33,36)(35,48)(40,42)(44,45);(10,14)(13,15)(17,22,24,23)(18,28,29,40)(19,32,34,33)(20,36,31,39)(21,37,38,41)(25,30,44,48)(26,45,35,43)(27,46,47,42)

Polycyclic presentation:

2^9|b^2=fgi,c^2=gh,d^2=gi,e^2=i,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, 6990598);