#7091411

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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)(19,27,30,37)(20,24,32,39)(21,34,35,25)(23,29,38,31)(28,41,43,48)(33,46,45,40)(36,44,47,42);(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,37)(26,38)(27,34)(31,35)(32,43)(33,44)(39,47)(40,48)(41,46)(42,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,22,32)(18,24,26,39)(19,28,30,43)(21,33,35,45)(23,36,38,47)(25,40,34,46)(27,41,37,48)(29,44,31,42);(11,16)(13,15)(19,29)(20,32)(21,35)(23,37)(24,39)(25,34)(27,38)(28,42)(30,31)(36,41)(43,44)(47,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,30)(20,32)(23,38)(24,39)(27,37)(29,31)(33,45)(40,46)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 6998607);