Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8507 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(8,11,12,13)(15,16,20,22,19,18,26,27)(17,21,25,29,24,23,28,30);(7,8)(9,13)(10,12)(11,14)(15,17,19,24)(16,21,18,23)(20,28,26,25)(22,30,27,29);(4,6)(9,14)(11,13)(15,18)(16,19)(17,23)(20,22)(21,24)(25,29)(26,27)(28,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94803);