#6587486

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(17,18,23,28)(19,24,32,41)(20,25,34,43)(21,26,31,40)(22,27,30,39)(29,38,47,48)(33,42,36,45)(35,44,37,46);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,33)(21,30)(22,31)(23,32)(25,42)(26,39)(27,40)(28,41)(29,35)(34,36)(37,47)(38,44)(43,45)(46,48);(10,13)(14,16)(17,20,23,34)(18,25,28,43)(19,29,32,47)(21,36,31,33)(22,37,30,35)(24,38,41,48)(26,45,40,42)(27,46,39,44);(9,11,12,15)(10,14,13,16)(17,21,23,31)(18,26,28,40)(19,30,32,22)(20,29,34,47)(24,39,41,27)(25,38,43,48)(33,35,36,37)(42,44,45,46);(1,4,5,3)(2,6,7,8)(19,31)(20,34)(21,32)(22,30)(24,40)(25,43)(26,41)(27,39)(29,36)(33,47)(38,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,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, 6494682);