Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#49561 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8442 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);(4,6)(15,16,21,25)(17,26,19,22)(18,23,24,20)(27,28,29,30);(8,11)(9,13)(15,17,21,19)(16,22,25,26)(18,29,24,27)(20,30,23,28);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,18,21,24)(16,23,25,20)(17,27,19,29)(22,30,26,28);(7,8)(9,12)(10,11)(13,14)(15,19,21,17)(16,22,25,26)(18,27,24,29)(20,30,23,28);(7,9)(8,12)(10,13)(11,14)(15,20,21,23)(16,24,25,18)(17,28,19,30)(22,29,26,27)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612728);