#6581259

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

1370 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,34,36,25)(21,37,35,26)(23,28,39,45)(30,41,38,44)(31,42,48,47)(33,46,40,43);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,38)(22,32)(23,40)(25,42)(26,44)(27,29)(28,46)(30,35)(33,39)(34,47)(36,48)(37,41)(43,45);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,22,36)(18,25,27,34)(19,30,32,38)(21,39,35,23)(24,41,29,44)(26,45,37,28)(31,33,48,40)(42,43,47,46);(1,4,5,3)(2,8,6,7)(20,35)(21,36)(23,39)(25,37)(26,34)(28,45)(30,31)(33,40)(38,48)(41,42)(43,46)(44,47);(17,21,22,35)(18,26,27,37)(19,31,32,48)(20,23,36,39)(24,42,29,47)(25,28,34,45)(30,40,38,33)(41,46,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6488455);