#6582367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6489563);