#8726654

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633850);