#7187826

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7095022);