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×#51904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7980 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);(3,4,5,6)(15,16)(17,21)(18,22)(19,20)(23,25)(24,29)(26,30)(27,28);(7,8,10,11)(9,13,14,12)(16,19)(17,23)(22,27)(24,26);(3,4)(5,6)(8,11)(12,13)(15,17,20,23)(16,21,19,25)(18,26,28,24)(22,30,27,29);(7,8)(9,12)(10,11)(13,14)(15,18,20,28)(16,22,19,27)(17,24,23,26)(21,29,25,30);(7,9,10,14)(8,12,11,13)(15,19,20,16)(17,25,23,21)(18,27,28,22)(24,30,26,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615152);