#6349421

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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

2753 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)(21,27)(22,26)(23,28)(24,31)(30,43)(32,38)(33,42)(34,41)(35,39)(36,40)(37,45)(44,48)(46,47);(1,3,6,13)(2,7,10,16)(4,12,5,14)(8,15,9,11)(17,19,23,33)(18,24,28,38)(20,30,36,46)(21,31,22,32)(25,37,40,48)(26,29,27,42)(34,44,35,45)(39,47,41,43);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,37)(26,41)(27,39)(28,40)(29,43)(31,45)(32,44)(33,46)(38,48)(42,47);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,29,38,42)(25,39,40,41)(30,44,46,45)(37,47,48,43);(3,13)(7,16)(11,15)(12,14)(17,21,23,22)(18,27,28,26)(19,32,33,31)(20,35,36,34)(24,29,38,42)(25,39,40,41)(30,44,46,45)(37,43,48,47)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256617);