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×#51202 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7764 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)(7,8)(9,13)(10,11)(12,14)(15,16,20,18)(17,21,24,25)(19,22,28,26)(23,29,27,30);(8,11)(12,13)(15,17,20,24)(16,21,18,25)(19,27,28,23)(22,30,26,29);(3,4,5,6)(15,17)(16,21)(18,25)(19,23)(20,24)(22,29)(26,30)(27,28);(7,8)(9,12)(10,11)(13,14)(15,18,20,16)(17,21,24,25)(19,26,28,22)(23,29,27,30);(7,9,10,14)(8,12,11,13)(15,19)(16,22)(17,23)(18,26)(20,28)(21,29)(24,27)(25,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614055);