#6351976

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

2599 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,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, 6259172);