Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
| direct product | C22×#2783 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(6,9)(7,11)(13,14,16,18)(15,19,20,17)(22,23)(26,27);(6,9)(10,12)(21,22,25,23)(24,27,28,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,25,22)(24,27,28,26);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,17,18,19)(21,24)(22,26)(23,27)(25,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540721);