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 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(18,21)(19,25)(24,28)(26,31)(33,34,39,41)(35,44,46,40)(36,47,42,37)(38,48,45,43);(1,3)(2,8)(4,5)(6,13)(7,11)(9,16)(10,15)(12,14)(17,18,22,21)(19,23,25,27)(20,24,29,28)(26,30,31,32)(33,35,39,46)(34,40,41,44)(36,48,42,43)(37,45,47,38);(2,9)(3,11)(4,14)(10,13)(19,25)(23,27)(26,31)(30,32)(34,41)(36,42)(40,44)(43,48);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,39,42)(34,37,41,47)(35,43,46,48)(38,44,45,40);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,20,22,29)(18,24,21,28)(19,26,25,31)(23,30,27,32)(33,37)(34,42)(35,45)(36,41)(38,46)(39,47)(40,43)(44,48);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,21,22,18)(19,27,25,23)(20,28,29,24)(26,32,31,30)(33,38,39,45)(34,43,41,48)(35,37,46,47)(36,44,42,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014433);