#6587415

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,47)(32,41)(33,45)(34,39)(35,43)(36,44)(37,40)(38,46)(42,48);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,33)(21,23)(22,31)(25,42)(26,40)(27,29)(28,41)(30,36)(32,37)(34,38)(35,48)(39,44)(43,45)(46,47);(10,13)(14,16)(17,20,22,35)(18,25,27,43)(19,30,31,34)(21,33,32,48)(23,38,37,36)(24,39,29,47)(26,45,41,42)(28,46,40,44);(9,11,12,15)(10,14,13,16)(17,21,22,32)(18,26,27,41)(19,23,31,37)(20,34,35,30)(24,40,29,28)(25,39,43,47)(33,38,48,36)(42,44,45,46);(1,4,5,3)(2,6,7,8)(19,21)(23,37)(24,41)(26,29)(28,40)(30,33)(31,32)(34,48)(36,38)(39,42)(44,46)(45,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,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, 6494611);