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×#44455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7718 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);(4,6)(7,8,11,13)(9,12,14,10)(15,16,20,23)(17,24,22,19)(18,21,28,30)(25,29,26,27);(7,9,11,14)(8,10,13,12)(17,22)(19,24)(25,26)(27,29);(3,4)(5,6)(15,17,20,22)(16,19,23,24)(18,26,28,25)(21,29,30,27);(7,9)(8,12)(10,13)(11,14)(15,18)(16,21)(17,25)(19,27)(20,28)(22,26)(23,30)(24,29);(7,10,11,12)(8,9,13,14)(15,19)(16,22)(17,23)(18,27)(20,24)(21,26)(25,30)(28,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576141);