#6351927

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259123);