#6581567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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

1319 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)(17,18,22,27)(19,29,32,24)(20,25,23,28)(21,26,34,40)(30,41,35,37)(31,43,46,38)(33,39,36,42)(44,48,45,47);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(19,30)(20,23)(21,33)(24,37)(25,28)(26,39)(29,41)(32,35)(34,36)(40,42)(44,45)(47,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,22,32)(18,24,27,29)(20,30,23,35)(21,31,34,46)(25,37,28,41)(26,38,40,43)(33,44,36,45)(39,47,42,48);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(21,34)(22,23)(26,40)(27,28)(30,35)(31,45)(37,41)(38,48)(43,47)(44,46);(17,21,22,34)(18,26,27,40)(19,31,32,46)(20,33,23,36)(24,38,29,43)(25,39,28,42)(30,44,35,45)(37,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488763);