#6582669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6489865);