#6630842

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,40)(31,39)(33,38)(35,42)(37,41)(43,46)(44,48)(45,47);(2,5)(6,8)(17,19,22,33,23,30,21,32)(18,24,26,40,28,38,27,29)(20,31,36,46,37,44,35,45)(25,39,34,47,42,48,41,43);(9,11,12,16)(10,13,14,15)(19,30)(24,38)(29,40)(31,44)(32,33)(39,48)(43,47)(45,46);(1,3,4,7)(2,6,5,8)(17,20,21,35,23,37,22,36)(18,25,27,41,28,42,26,34)(19,31,32,45,30,44,33,46)(24,39,29,43,38,48,40,47);(17,21,23,22)(18,26,28,27)(19,32,30,33)(20,35,37,36)(24,40,38,29)(25,34,42,41)(31,45,44,46)(39,47,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538038);