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 | |
4096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18,23,21)(19,27,22,24)(20,26,30,25)(28,31,29,32)(33,34,40,44)(35,45,37,41)(36,42,46,47)(38,39,43,48);(1,2,6,7)(3,4,10,9)(5,8,15,13)(11,12,16,14)(17,19,23,22)(18,24,21,27)(20,28,30,29)(25,31,26,32)(33,35,40,37)(34,41,44,45)(36,38,46,43)(39,42,48,47);(2,7)(3,10)(8,13)(11,16)(17,20,23,30)(18,25,21,26)(19,28,22,29)(24,31,27,32)(33,36,40,46)(34,42,44,47)(35,38,37,43)(39,41,48,45);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20)(18,26)(19,28)(21,25)(22,29)(23,30)(24,32)(27,31)(33,37,40,35)(34,41,44,45)(36,38,46,43)(39,47,48,42);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21,23,18)(19,24,22,27)(20,26,30,25)(28,31,29,32)(33,38)(34,39)(35,46)(36,37)(40,43)(41,42)(44,48)(45,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22,23,19)(18,27,21,24)(20,29,30,28)(25,32,26,31)(33,39)(34,43)(35,42)(36,41)(37,47)(38,44)(40,48)(45,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113723);