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×#37711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7416 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);(7,8,11,9)(10,13,14,12)(15,16)(17,22)(18,26)(19,24)(20,21)(23,28)(25,27)(29,30);(4,6)(7,9,11,8)(10,13,14,12)(16,21)(17,24)(23,26)(25,30);(3,4)(5,6)(7,10,11,14)(8,12,9,13)(15,17,20,24)(16,22,21,19)(18,25,28,30)(23,29,26,27);(15,18,20,28)(16,23,21,26)(17,25,24,30)(19,27,22,29);(3,4)(5,6)(15,19,20,22)(16,24,21,17)(18,27,28,29)(23,30,26,25)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542322);