#6350860

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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

2633 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)(3,9)(4,14)(5,10)(6,16)(7,11)(8,12)(13,15)(17,18,23,27)(19,28,31,38)(20,25,32,40)(21,26,22,35)(24,36,39,30)(29,42,43,48)(33,41,34,45)(37,46,47,44);(1,3,7,13)(2,8,11,14)(4,5,12,6)(9,10,15,16)(17,19,23,31)(18,24,27,39)(20,29,32,43)(21,30,22,36)(25,37,40,47)(26,38,35,28)(33,44,34,46)(41,48,45,42);(1,4)(2,9)(3,5)(6,13)(7,12)(8,10)(11,15)(14,16)(17,20,23,32)(18,25,27,40)(19,29,31,43)(21,33,22,34)(24,37,39,47)(26,41,35,45)(28,42,38,48)(30,44,36,46);(1,2)(3,8)(4,15)(5,16)(6,10)(7,11)(9,12)(13,14)(20,32)(25,40)(29,43)(33,34)(37,47)(41,45)(42,48)(44,46);(1,5,7,6)(2,10,11,16)(3,12,13,4)(8,15,14,9)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,32,34)(24,38,39,28)(25,41,40,45)(29,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258056);