#6350593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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

2720 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,30)(20,34)(21,36)(22,27)(23,28)(24,29)(25,33)(26,39)(31,45)(32,41)(35,44)(37,43)(38,42)(40,47)(46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,24,20)(18,25,29,41)(21,31,39,35)(22,32,23,33)(26,40,36,48)(27,30,28,34)(37,46,38,47)(42,45,43,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,24,19)(18,25,29,41)(21,35,39,31)(22,33,23,32)(26,40,36,48)(27,30,28,34)(37,47,38,46)(42,45,43,44);(2,9)(7,16)(8,13)(10,15)(17,21)(18,26)(19,31)(20,35)(22,38)(23,37)(24,39)(25,40)(27,43)(28,42)(29,36)(30,44)(32,47)(33,46)(34,45)(41,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,22,24,23)(18,27,29,28)(19,32,20,33)(21,37,39,38)(25,30,41,34)(26,42,36,43)(31,46,35,47)(40,45,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257789);