#6558484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1731 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)(17,18)(19,24)(20,32)(21,26)(22,37)(23,27)(25,35)(28,43)(29,39)(30,40)(31,46)(33,47)(34,42)(36,48)(38,45)(41,44);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,29)(25,41)(26,40)(27,39)(28,36)(32,47)(34,45)(35,44)(37,46)(38,42)(43,48);(10,13)(14,16)(17,20,21,34,23,36,22,35)(18,25,37,48,27,42,26,32)(19,28,30,44,29,33,31,45)(24,38,46,47,39,41,40,43);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,29)(24,39)(28,33)(30,31)(38,41)(40,46)(43,47)(44,45);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,31)(20,34,36,35)(24,40,39,46)(25,32,42,48)(28,44,33,45)(38,43,41,47)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^d=ehi

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, 6465680);