Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#40532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7690 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)(15,16,20,19)(17,23,25,21)(18,26,28,22)(24,29,27,30);(7,8)(9,13)(10,11)(12,14)(16,19)(21,23)(22,26)(29,30);(3,4)(5,6)(8,11)(12,13)(15,17,20,25)(16,21,19,23)(18,27,28,24)(22,30,26,29);(7,8)(9,12)(10,11)(13,14)(15,18,20,28)(16,22,19,26)(17,24,25,27)(21,29,23,30);(7,9)(8,12)(10,14)(11,13)(15,19,20,16)(17,23,25,21)(18,26,28,22)(24,30,27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546589);