#6582667

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=f,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, 6489863);