Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#20639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,6)(7,8,11,16,12,15,24,29)(9,13,19,27,18,26,34,37)(10,20,21,30,22,14,23,28)(17,31,33,38,32,25,35,36);(3,4,5,6)(7,9)(8,13)(10,17)(11,19)(12,18)(14,25)(15,26)(16,27)(20,31)(21,33)(22,32)(23,35)(24,34)(28,36)(29,37)(30,38);(7,10,12,22)(8,14,15,20)(9,17,18,32)(11,23,24,21)(13,25,26,31)(16,30,29,28)(19,35,34,33)(27,38,37,36);(9,18)(13,26)(17,32)(19,34)(25,31)(27,37)(33,35)(36,38)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277975);