#6351301

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

2716 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,30)(20,34)(21,26)(22,29)(23,28)(24,27)(25,32)(31,45)(33,40)(35,48)(36,44)(37,43)(38,42)(39,46)(41,47);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(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)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(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);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(17,21,24,36)(18,26,27,44)(19,31,20,35)(22,37,23,38)(25,39,40,41)(28,42,29,43)(30,45,34,48)(32,46,33,47);(2,10)(7,16)(8,15)(9,14)(18,27)(21,36)(25,40)(28,29)(30,34)(31,35)(37,38)(46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^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, 6258497);