#6350810

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

2606 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)(3,12)(4,15)(5,9)(6,10)(7,11)(8,14)(13,16)(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);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,12,10,15)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,43)(33,44)(34,46)(40,47)(41,48)(42,45);(1,4,7,3)(2,8,11,16)(5,14,6,13)(9,12,10,15)(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)(4,16)(5,10)(6,9)(7,11)(12,14)(13,15)(20,32)(25,40)(29,43)(33,34)(37,47)(41,45)(42,48)(44,46);(1,5,7,6)(2,9,11,10)(3,13,4,14)(8,12,16,15)(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=i,c^2=h,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, 6258006);