Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,13)(10,11)(12,14)(16,20)(17,23)(18,26)(28,30);(8,11)(12,13)(15,16,19,20)(17,24,23,21)(18,22,26,27)(25,30,29,28);(15,17,19,23)(16,21,20,24)(18,25,26,29)(22,28,27,30);(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,27)(29,30);(7,9)(8,12)(10,14)(11,13)(15,18)(16,22)(17,25)(19,26)(20,27)(21,28)(23,29)(24,30)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533137);