Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37783 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(8,11)(12,14)(15,16,21,20)(17,24,19,22)(18,26,28,23)(25,29,27,30);(4,6)(8,11)(9,13)(15,17,21,19)(16,22,20,24)(18,27,28,25)(23,30,26,29);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,18,21,28)(16,23,20,26)(17,25,19,27)(22,29,24,30);(15,19,21,17)(16,24,20,22)(18,27,28,25)(23,30,26,29);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,20,21,16)(17,24,19,22)(18,26,28,23)(25,30,27,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542227);