#6466950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,25)(21,35)(22,28)(23,27)(26,34)(29,39)(30,43)(31,42)(32,41)(33,44)(36,48)(37,45)(38,46)(40,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,36)(22,38)(23,32)(25,43)(26,39)(27,41)(28,46)(29,34)(31,37)(33,47)(35,48)(40,44)(42,45);(9,12,13,11)(10,14,15,16)(17,20,22,34,23,33,37,21)(18,25,28,26,27,44,45,35)(19,29,31,30,32,36,38,47)(24,39,42,43,41,48,46,40);(17,21,37,33,23,34,22,20)(18,26,45,25,27,35,28,44)(19,30,38,29,32,47,31,36)(24,40,46,48,41,43,42,39);(1,3,4,7)(2,6,5,8)(18,27)(20,33)(21,34)(24,41)(28,45)(29,36)(30,47)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374146);