Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#20459 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3050 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);(3,4,6,5)(7,8,11,16,13,18,12,17)(9,19,21,28,22,32,26,14)(10,15,24,30,23,29,25,31)(20,33,35,37,34,38,36,27);(7,9,12,21,13,22,11,26)(8,14,17,19,18,28,16,32)(10,20,25,35,23,34,24,36)(15,27,31,33,29,37,30,38);(3,5,6,4)(7,10,13,23)(8,15,18,29)(9,20,22,34)(11,24,12,25)(14,27,28,37)(16,30,17,31)(19,33,32,38)(21,35,26,36);(10,23)(15,29)(20,34)(24,25)(27,37)(30,31)(33,38)(35,36)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277121);