#6351978

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

2657 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,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, 6259174);