#6350367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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

2593 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,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, 6257563);