#9027322

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

4086 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:

(2,7)(3,8)(9,14)(10,15)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34,39,44)(35,45,46,40)(36,41,43,38)(37,42,47,48);(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(35,46)(37,47)(40,45)(42,48);(3,8)(4,11)(5,13)(9,14)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,35,39,46)(34,40,44,45)(36,37,43,47)(38,42,41,48);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,36,39,43)(34,41,44,38)(35,47,46,37)(40,42,45,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,37,39,47)(34,42,44,48)(35,36,46,43)(38,45,41,40);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(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=i,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, 8934518);