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×#39129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8184 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)(17,21)(18,25)(19,20)(22,26)(23,29)(24,27)(28,30);(7,8,10,12)(9,13,14,11)(16,20)(21,27)(22,25)(28,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17,19,24)(16,21,20,27)(18,23,26,30)(22,28,25,29);(15,18)(16,22)(17,23)(19,26)(20,25)(21,28)(24,30)(27,29);(7,9,10,14)(8,11,12,13)(15,16)(17,21)(18,22)(19,20)(23,28)(24,27)(25,26)(29,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544130);