#9027326

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
prime radical	#47629	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934522);