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×#37811 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7140 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);(8,12)(13,14)(15,16,19,22)(17,23,25,20)(18,21,26,28)(24,29,30,27);(4,6)(15,17,19,25)(16,20,22,23)(18,24,26,30)(21,27,28,29);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,20)(18,24)(19,25)(21,27)(22,23)(26,30)(28,29);(7,9,11,10)(8,13,12,14)(15,16)(17,20)(18,21)(19,22)(23,25)(24,27)(26,28)(29,30);(3,4)(5,6)(7,10,11,9)(8,14,12,13)(15,18,19,26)(16,21,22,28)(17,24,25,30)(20,27,23,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542234);