Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(4,8)(9,15)(10,16)(17,18)(19,20)(21,29)(22,23)(24,26)(25,28)(27,34)(30,31)(32,33);(19,21,25,24)(20,26,28,29)(22,32,27,30)(23,31,34,33);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,22,25,27)(20,23,28,34)(21,30,24,32)(26,33,29,31);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(19,23)(20,27)(21,31)(22,28)(24,33)(25,34)(26,30)(29,32);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18)(19,24,25,21)(20,26,28,29)(22,32,27,30)(23,33,34,31);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493489);