#6349885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2851 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,43)(30,39)(32,38)(33,40)(34,41)(35,42)(37,45)(44,47)(46,48);(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,37)(26,38)(27,39)(28,36)(33,45)(34,44)(35,46)(40,48)(41,43)(42,47);(1,3,6,13)(2,8,10,16)(4,12,5,14)(7,15,9,11)(17,20,23,35)(18,25,27,42)(19,29,32,46)(21,33,22,34)(24,37,39,47)(26,40,36,41)(28,43,38,48)(30,44,31,45);(1,4,6,5)(2,9,10,7)(3,12,13,14)(8,11,16,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,28,39,38)(25,40,42,41)(29,44,46,45)(37,43,47,48);(3,13)(8,16)(11,15)(12,14)(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,33)(24,38,39,28)(25,41,42,40)(29,44,46,45)(37,43,47,48)

Polycyclic presentation:

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

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, 6257081);