Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(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)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(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)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(35,46)(37,47)(40,45)(42,48);(1,3,6,11)(2,7,9,5)(4,12,14,10)(8,13,16,15)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(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)(18,19)(21,29)(22,26)(23,25)(24,31)(27,32)(28,30)(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)(17,21,22,30)(18,24,25,32)(19,27,23,31)(20,28,26,29)(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)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,38,39,41)(34,43,44,36)(35,48,46,42)(37,45,47,40)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=h,e^2=gh,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=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8934536);