#8633858

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

4180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8541054);