#6351213

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

2623 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 6258409);