Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×#20593 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
16994 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)(5,6)(7,8,11,16,12,15,24,29)(9,17,19,30,20,13,23,26)(10,14,22,28,21,27,34,37)(18,31,33,38,32,25,35,36);(7,9,12,20)(8,13,15,17)(10,18,21,32)(11,23,24,19)(14,25,27,31)(16,30,29,26)(22,35,34,33)(28,38,37,36);(4,6)(7,10)(8,14)(9,18)(11,22)(12,21)(13,25)(15,27)(16,28)(17,31)(19,33)(20,32)(23,35)(24,34)(26,36)(29,37)(30,38);(10,21)(14,27)(18,32)(22,34)(25,31)(28,37)(33,35)(36,38)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277898);