#6818716

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1198 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fghi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6725912);