#6558780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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)(4,6)(5,8)(17,18,23,27)(19,24,30,41)(20,31,33,47)(21,26,22,37)(25,34,43,35)(28,44,32,46)(29,39,36,40)(38,48,42,45);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,32)(21,36)(22,29)(23,30)(25,42)(26,40)(27,41)(28,33)(31,46)(34,45)(35,48)(37,39)(38,43)(44,47);(10,13)(14,16)(17,20,21,34,23,33,22,35)(18,25,37,47,27,43,26,31)(19,28,36,48,30,32,29,45)(24,38,39,46,41,42,40,44);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,33)(25,43)(28,32)(31,47)(34,35)(38,42)(44,46)(45,48);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,34,33,35)(24,39,41,40)(25,31,43,47)(28,45,32,48)(38,46,42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465976);