#6351320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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

2578 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258516);