#6351928

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

2644 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)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18)(19,31)(20,37)(21,27)(22,28)(23,29)(24,30)(25,35)(26,38)(32,42)(33,45)(34,44)(36,48)(39,46)(40,47)(41,43);(1,3,7,13)(2,8,4,11)(5,12,6,15)(9,16,14,10)(17,19)(18,25)(20,32)(21,33)(22,34)(23,35)(24,36)(26,42)(27,43)(28,31)(29,48)(30,44)(37,46)(38,40)(39,41)(45,47);(1,2,7,4)(3,8,13,11)(5,9,6,14)(10,12,16,15)(17,20,24,33)(18,26,30,43)(19,32,36,21)(22,38,23,41)(25,42,44,27)(28,45,29,37)(31,47,48,46)(34,40,35,39);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,21,24,32)(18,27,30,42)(19,33,36,20)(22,40,23,39)(25,43,44,26)(28,47,29,46)(31,45,48,37)(34,38,35,41);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,33,41)(21,39,32,40)(25,31,44,48)(26,45,43,37)(27,46,42,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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, 6259124);