#6349751

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

2701 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=i,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, 6256947);