#6351900

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

2776 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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, 6259096);