#7092666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1274 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)(17,18,21,26,22,27,35,41)(19,28,31,40,30,24,38,23)(20,32,36,46,37,25,43,45)(29,42,33,39,34,44,47,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,40)(20,33,37,47)(21,38,35,31)(24,41,28,26)(25,44,32,42)(29,43,34,36)(39,46,48,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,34)(21,31)(22,30)(23,26)(25,44)(27,28)(29,37)(32,42)(33,43)(35,38)(36,47)(39,46)(40,41)(45,48);(11,16)(13,15)(17,20,35,43,22,37,21,36)(18,25,41,46,27,32,26,45)(19,29,38,47,30,34,31,33)(23,39,24,42,40,48,28,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,30)(23,40)(24,28)(29,34)(31,38)(33,47)(39,48)(42,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=gh,d^2=fi,e^2=gi,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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6999862);