#8630851

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

4222 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,23,25)(19,24,29,31)(20,26,27,21)(22,28,32,30)(33,34)(35,45)(36,42)(37,39)(38,43)(40,44)(41,48)(46,47);(2,4)(3,10)(8,12)(11,13)(17,19)(18,24)(20,30)(21,32)(22,26)(23,29)(25,31)(27,28)(33,35,40,48)(34,41,44,45)(36,46,37,43)(38,39,47,42);(2,4)(7,9)(8,12)(15,16)(18,25)(19,29)(21,26)(28,30)(33,36,40,37)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,20,23,27)(18,26,25,21)(19,28,29,30)(22,24,32,31)(33,37,40,36)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21,23,26)(18,27,25,20)(19,22,29,32)(24,30,31,28)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,41,37,45);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22)(18,28)(19,26)(20,31)(21,29)(23,32)(24,27)(25,30)(33,39,40,42)(34,37,44,36)(35,47,48,38)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gh,e^2=gi,f^2=g,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, 8538047);