#7083486

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=ghi,d^2=gi,e^2=i,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, 6990682);