#6851469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1231 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

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, 6758665);