Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32144 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);(3,4,10,9)(5,11,14,7)(6,15,8,12)(13,17,16,18)(19,20)(21,29)(22,32)(23,26)(24,28)(25,27)(30,31)(33,34);(3,5)(4,11)(6,16)(7,9)(8,13)(10,14)(12,17)(15,18)(19,21,26,25)(20,27,23,29)(22,30,28,33)(24,31,32,34);(3,6,10,8)(4,12,9,15)(5,13,14,16)(7,17,11,18)(19,22)(20,24)(21,30)(23,32)(25,33)(26,28)(27,31)(29,34);(3,7)(4,5)(6,17)(8,18)(9,14)(10,11)(12,13)(15,16)(19,23,26,20)(21,27,25,29)(22,32,28,24)(30,31,33,34);(3,8,10,6)(4,12,9,15)(5,13,14,16)(7,18,11,17)(19,24,26,32)(20,28,23,22)(21,31,25,34)(27,33,29,30);(3,9,10,4)(5,7,14,11)(6,15,8,12)(13,17,16,18)(19,25,26,21)(20,27,23,29)(22,33,28,30)(24,31,32,34)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493594);