#6571698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2934 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)(19,24)(20,25)(21,35)(22,26)(23,27)(28,41)(29,45)(30,37)(31,38)(32,46)(33,39)(34,40)(36,44)(42,47)(43,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,36)(26,37)(27,38)(32,42)(33,43)(34,44)(35,45)(39,47)(40,41)(46,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,25)(19,28)(21,32)(22,33)(23,34)(24,36)(26,39)(27,40)(29,42)(30,43)(31,44)(35,46)(37,47)(38,41)(45,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,23,22)(18,26,27,35)(19,29,31,30)(20,32,34,33)(24,37,38,45)(25,39,40,46)(28,42,44,43)(36,47,41,48);(4,12)(7,14)(10,16)(13,15)(17,22,23,21)(18,26,27,35)(19,30,31,29)(20,33,34,32)(24,37,38,45)(25,39,40,46)(28,43,44,42)(36,47,41,48)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6478894);