#6353485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,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, 6260681);