#6851501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1226 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)(19,24,33,41)(20,26,35,43)(21,27,22,37)(25,36,44,34)(29,45,30,39)(31,40,32,48)(38,47,42,46);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,15,12,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,42)(26,39)(27,40)(28,41)(29,35)(34,47)(36,46)(37,48)(38,44)(43,45);(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,43)(28,44)(30,33)(31,46)(32,47)(39,40)(41,42)(45,48);(1,3,4,7)(2,6,5,8)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,39)(25,37)(27,44)(28,43)(29,33)(31,46)(32,47)(38,40)(41,45)(42,48);(2,5)(6,8)(9,12,13,11)(10,15,16,14)(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,34,35,36)(24,40,41,48)(25,43,44,26)(29,46,30,47)(38,39,42,45)

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=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6758697);