#6571688

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6478884);