Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#41951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7616 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)(15,16,21,20)(17,24,18,22)(19,27,28,23)(25,29,26,30);(7,8)(9,10)(11,12)(13,14)(15,17,21,18)(16,22,20,24)(19,25,28,26)(23,29,27,30);(3,4)(5,6)(8,12)(9,13)(15,18,21,17)(16,22,20,24)(19,25,28,26)(23,30,27,29);(7,9,11,13)(8,10,12,14)(15,19,21,28)(16,23,20,27)(17,25,18,26)(22,29,24,30);(7,10)(8,13)(9,12)(11,14)(15,20,21,16)(17,22,18,24)(19,27,28,23)(25,29,26,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547712);