Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#44461 |
| 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 | |
7286 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);(3,4,6,5)(7,8,11,9)(10,13,14,12)(15,16)(17,22)(18,23)(19,26)(20,21)(24,29)(25,28)(27,30);(4,5)(7,9,11,8)(10,13,14,12)(16,21)(17,25)(18,27)(19,29);(7,10)(8,12)(9,13)(11,14)(15,17,20,25)(16,22,21,28)(18,24,27,26)(19,23,29,30);(3,5,6,4)(15,18,20,27)(16,23,21,30)(17,26,25,24)(19,28,29,22);(15,19)(16,24)(17,23)(18,28)(20,29)(21,26)(22,27)(25,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576004);