#6463669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,38,22,27,21,26)(19,28,31,24,30,41,32,40)(20,33,35,44,37,25,36,43)(29,46,48,42,34,45,47,39);(2,5)(6,7)(10,14)(11,12)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,42)(26,41)(27,40)(28,38)(29,37)(33,39)(35,48)(36,47)(43,46)(44,45);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,37)(18,25,27,33)(19,29,30,34)(21,36,23,35)(24,39,40,42)(26,44,38,43)(28,45,41,46)(31,48,32,47);(9,12,13,11)(10,15,14,16)(19,30)(24,40)(28,41)(29,34)(31,32)(39,42)(45,46)(47,48);(17,21,22,23)(18,26,27,38)(19,31,30,32)(20,35,37,36)(24,41,40,28)(25,43,33,44)(29,47,34,48)(39,45,42,46)

Polycyclic presentation:

2^9|a^2=fg,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370865);