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×#51411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7378 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)(7,8)(9,12)(10,11)(13,14)(15,16,20,23)(17,21,24,18)(19,22,27,29)(25,28,30,26);(8,10)(12,13)(17,24)(18,21)(19,27)(22,29);(3,4)(5,6)(7,8,11,10)(9,13,14,12)(15,17,20,24)(16,21,23,18)(19,25,27,30)(22,28,29,26);(7,9,11,14)(8,12,10,13)(15,18,20,21)(16,17,23,24)(19,26,27,28)(22,25,29,30);(7,10,11,8)(9,13,14,12)(15,19)(16,22)(17,25)(18,26)(20,27)(21,28)(23,29)(24,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614220);