Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×#12551 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1698 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);(7,8,10,15,11,16,22,30,12,14,23,27,24,28,35,37)(9,17,19,29,20,31,33,38,18,13,21,25,32,26,34,36);(3,4,5,6)(7,9)(8,13)(10,21)(11,20)(12,18)(14,17)(15,29)(16,26)(19,23)(22,34)(24,32)(25,27)(28,31)(30,38)(33,35)(36,37);(9,18)(13,17)(19,21)(20,32)(25,29)(26,31)(33,34)(36,38)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400043);