#6829728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,24,30,37)(20,25,32,39)(21,26,34,41)(22,27,35,42)(29,36,44,47)(31,38,45,48)(33,40,43,46);(1,3)(2,5)(4,8)(6,7)(17,19,23,30)(18,24,28,37)(20,31,32,45)(21,22,34,35)(25,38,39,48)(26,27,41,42)(29,43,44,33)(36,46,47,40);(10,13)(14,16)(17,20,22,33,23,32,35,43)(18,25,27,40,28,39,42,46)(19,29,21,31,30,44,34,45)(24,36,26,38,37,47,41,48);(2,6)(5,7)(20,32)(25,39)(29,44)(31,45)(33,43)(36,47)(38,48)(40,46);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,30,23,34,35,19)(18,26,27,37,28,41,42,24)(20,31,33,44,32,45,43,29)(25,38,40,47,39,48,46,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6736924);