Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63037 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,17,21)(13,19,14,18)(15,16,20,26)(22,25,24,23);(4,7)(8,9)(11,13,17,14)(12,18,21,19)(15,22,20,24)(16,23,26,25);(3,4)(5,8)(6,7)(9,10)(11,13,17,14)(12,18,21,19)(15,22,20,24)(16,23,26,25);(3,5,6,10)(4,8,7,9)(11,14,17,13)(12,19,21,18)(15,24,20,22)(16,25,26,23);(11,15)(12,16)(13,22)(14,24)(17,20)(18,23)(19,25)(21,26);(11,16)(12,20)(13,23)(14,25)(15,21)(17,26)(18,24)(19,22)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493555);