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×#62941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(3,4,10,14)(5,15,18,11)(6,7,8,12)(9,16,17,13)(21,27)(25,29)(28,33)(32,34);(3,5,10,18)(4,11,14,15)(6,13,8,16)(7,17,12,9)(19,20,24,23)(21,25,27,29)(22,26,30,31)(28,32,33,34);(3,6,10,8)(4,12,14,7)(5,16,18,13)(9,11,17,15)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,33)(31,34);(3,7)(4,6)(5,17)(8,14)(9,18)(10,12)(11,13)(15,16)(19,22)(20,26)(21,28)(23,31)(24,30)(25,32)(27,33)(29,34);(3,8,10,6)(4,12,14,7)(5,16,18,13)(9,15,17,11)(19,23,24,20)(21,29,27,25)(22,26,30,31)(28,32,33,34);(3,9,10,17)(4,13,14,16)(5,7,18,12)(6,11,8,15)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,33)(31,34)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493480);