#6560138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=i,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, 6467334);