Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31466 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)(5,9)(6,13)(7,8)(10,14)(11,15)(12,18)(16,17)(20,24)(22,30)(25,27)(28,34);(5,11)(9,15)(12,17)(16,18)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(4,8)(9,15)(10,13)(16,18)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,4)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14)(17,18)(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34);(3,5,7,11)(4,9,8,15)(6,12,14,17)(10,16,13,18);(3,6)(4,10)(5,12)(7,14)(8,13)(9,16)(11,17)(15,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493407);