#6352858

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352701
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260054);