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×#62945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31808 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);(3,4,8,12)(5,13,16,9)(6,17,15,10)(7,11,14,18)(20,24)(25,31)(26,32)(33,34);(3,5)(4,9)(6,14)(7,15)(8,16)(10,18)(11,17)(12,13);(3,6)(4,10)(5,7)(8,15)(9,11)(12,17)(13,18)(14,16)(19,20)(21,25)(22,26)(23,24)(27,33)(28,31)(29,34)(30,32);(3,6)(4,10)(5,14)(7,16)(8,15)(9,18)(11,13)(12,17)(19,21,23,28)(20,25,24,31)(22,29,30,27)(26,34,32,33);(3,7,8,14)(4,11,12,18)(5,15,16,6)(9,17,13,10)(19,22,23,30)(20,26,24,32)(21,27,28,29)(25,33,31,34);(3,4)(5,9)(6,10)(7,11)(8,12)(13,16)(14,18)(15,17)(19,20)(21,25)(22,26)(23,24)(27,33)(28,31)(29,34)(30,32)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493485);