Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11230 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1778 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);(7,8,12,16,13,18,11,17)(9,19,22,30,23,29,21,14)(10,24,25,32,26,31,27,15)(20,28,34,37,35,38,36,33);(3,4)(5,6)(7,9,11,21,13,23,12,22)(8,14,16,19,18,30,17,29)(10,20,25,34,26,35,27,36)(15,28,31,33,32,38,24,37);(4,6)(7,10,11,25,13,26,12,27)(8,15,16,31,18,32,17,24)(9,20,21,34,23,35,22,36)(14,28,19,33,30,38,29,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265228);