#7092701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1277 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)(17,18,23,38,22,28,21,27)(19,25,37,24,30,43,31,41)(20,32,39,45,35,26,36,46)(29,44,47,48,34,42,33,40);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,24,28,41)(20,33,35,47)(21,37,23,31)(25,27,43,38)(26,42,32,44)(29,39,34,36)(40,46,48,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,34)(21,31)(22,30)(23,37)(24,38)(26,44)(27,41)(28,43)(29,35)(32,42)(33,39)(36,47)(40,45)(46,48);(11,16)(13,15)(17,20,23,39,22,35,21,36)(18,26,38,46,28,32,27,45)(19,29,37,47,30,34,31,33)(24,40,43,44,41,48,25,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,30)(20,35)(24,41)(25,43)(26,32)(31,37)(36,39)(45,46)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=ghi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6999897);