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×#62952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32030 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)(21,30)(22,31)(23,24)(25,32)(26,29)(27,34)(28,33);(19,21,26,24)(20,23,29,30)(22,32,34,28)(25,31,33,27);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,22)(20,27)(21,28)(23,33)(24,32)(25,30)(26,34)(29,31);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(19,23,26,30)(20,21,29,24)(22,33,34,25)(27,28,31,32);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18)(19,24,26,21)(20,23,29,30)(22,32,34,28)(25,27,33,31);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,25,26,33)(20,28,29,32)(21,31,24,27)(22,30,34,23)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493492);