#6349696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2786 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)(3,11)(4,9)(5,8)(6,10)(7,14)(12,16)(13,15)(17,18)(19,29)(20,25)(21,28)(22,26)(23,27)(24,31)(30,43)(32,42)(33,41)(34,40)(35,39)(36,38)(37,46)(44,48)(45,47);(1,3,6,13)(2,7,10,16)(4,12,5,14)(8,15,9,11)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,38)(28,42)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,41)(27,40)(28,39)(29,43)(31,46)(32,45)(36,44)(38,48)(42,47);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,33,34,35)(24,29,38,42)(25,39,40,41)(30,44,45,46)(37,47,48,43);(2,10)(3,13)(8,9)(12,14)(18,27)(19,32)(25,40)(26,28)(30,45)(31,36)(39,41)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256892);