#6351170

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

2707 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^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, 6258366);