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×#34504 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7702 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);(3,4,6,5)(7,8,12,13)(9,11,10,14)(16,20)(17,23)(22,27)(24,29);(8,13)(9,10)(16,20)(21,26)(22,27)(28,30);(7,9,12,10)(8,14,13,11)(15,16)(17,21)(18,22)(19,20)(23,26)(24,28)(25,27)(29,30);(3,5,6,4)(7,10,12,9)(8,11,13,14)(15,17)(16,21)(18,24)(19,23)(20,26)(22,28)(25,29)(27,30);(7,11)(8,9)(10,13)(12,14)(15,18,19,25)(16,22,20,27)(17,24,23,29)(21,28,26,30)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533300);