#7101341

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=gi,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,f^b=fi,d^c=dhi,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, 7008537);