Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50645 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7178 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);(4,6)(15,16,20,23)(17,19,25,21)(18,22,28,30)(24,27,26,29);(7,8)(9,11)(10,13)(12,14)(15,17,20,25)(16,21,23,19)(18,24,28,26)(22,29,30,27);(3,4)(5,6)(8,9)(12,13)(15,17)(16,21)(18,26)(19,23)(20,25)(22,27)(24,28)(29,30);(7,9,11,8)(10,13,14,12)(15,18)(16,22)(17,24)(19,27)(20,28)(21,29)(23,30)(25,26);(7,10)(8,12)(9,13)(11,14)(15,19,20,21)(16,17,23,25)(18,27,28,29)(22,24,30,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613696);