#8630840

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

4191 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)(3,10,5,7)(4,8,13,16)(11,15,12,14)(17,18,23,25)(19,24,29,31)(20,26,27,21)(22,28,32,30)(33,34,40,46)(35,47,48,41)(36,43,37,42)(38,39,45,44);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,19)(18,24)(20,30)(21,32)(22,26)(23,29)(25,31)(27,28)(33,35)(34,41)(36,44)(37,39)(38,42)(40,48)(43,45)(46,47);(4,13)(8,16)(11,12)(14,15)(18,25)(19,29)(21,26)(28,30)(33,36,40,37)(34,42,46,43)(35,44,48,39)(38,47,45,41);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,20,23,27)(18,26,25,21)(19,28,29,30)(22,24,32,31)(33,37,40,36)(34,43,46,42)(35,44,48,39)(38,47,45,41);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,21,23,26)(18,27,25,20)(19,22,29,32)(24,30,31,28)(33,38,40,45)(34,44,46,39)(35,43,48,42)(36,41,37,47);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,22)(18,28)(19,26)(20,31)(21,29)(23,32)(24,27)(25,30)(33,39,40,44)(34,45,46,38)(35,37,48,36)(41,43,47,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=h,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, 8538036);