#8596419

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,e^2=gh,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, 8503615);