Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#12596 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,12,17,11,19,27,34,13,18,29,15,28,16,10,25)(9,20,23,35,22,14,21,30,24,32,36,38,37,31,26,33);(3,4,5,6)(7,9)(8,14,18,31)(10,26)(11,22)(12,23)(13,24)(15,33,17,30)(16,20,19,32)(21,27)(25,35,34,38)(28,37)(29,36);(7,10,11,12,13,27,28,29)(8,15,16,34,18,17,19,25)(9,21,22,36,24,26,37,23)(14,30,20,35,31,33,32,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400121);