#7091893

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(17,18,23,28)(19,29,32,48)(20,25,31,44)(21,26,30,46)(22,37,39,27)(24,40,43,33)(34,42,35,45)(36,47,38,41);(9,10)(11,15)(12,14)(13,16)(17,19,22,33,23,32,39,40)(18,24,27,29,28,43,37,48)(20,30,34,38,31,21,35,36)(25,41,45,46,44,47,42,26);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,31)(18,25,28,44)(19,30,32,21)(22,34,39,35)(24,41,43,47)(26,29,46,48)(27,45,37,42)(33,38,40,36);(11,16)(13,15)(17,21,23,30)(18,26,28,46)(19,31,32,20)(22,38,39,36)(24,42,43,45)(25,29,44,48)(27,47,37,41)(33,34,40,35);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,32)(20,31)(24,43)(25,44)(29,48)(33,40)(34,35)(42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=ghi,d^2=i,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,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, 6999089);