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×#42178 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7668 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)(9,14)(10,12)(11,13)(15,16,20,18)(17,21,23,25)(19,22,27,26)(24,28,29,30);(7,8,12,10)(9,13,11,14)(17,23)(19,27)(21,25)(22,26);(3,4,5,6)(7,9,12,11)(8,13,10,14)(15,17)(16,21)(18,25)(19,24)(20,23)(22,28)(26,30)(27,29);(7,10,12,8)(9,13,11,14)(15,18,20,16)(17,21,23,25)(19,26,27,22)(24,28,29,30);(7,11,12,9)(8,13,10,14)(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547892);