Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C22×#2772 |
| 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);(5,6)(7,11)(8,9)(10,12)(14,15)(18,19)(21,22,25,28)(23,26,27,24);(6,9)(7,12)(13,14,17,15)(16,19,20,18)(21,23,25,27)(22,26,28,24);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,24,25,26)(22,27,28,23);(5,7,8,12)(6,10,9,11)(13,16,17,20)(14,18,15,19)(21,22)(23,26)(24,27)(25,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540751);