Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C22×#2533 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,19,16,18)(23,27)(26,28);(5,6,8,10)(7,9,11,12)(13,15,17,16)(14,18,20,19)(21,22,25,24)(23,26,27,28);(5,6)(7,9)(8,10)(11,12)(13,16,17,15)(14,19,20,18)(21,23)(22,26)(24,28)(25,27);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,19)(17,20)(21,24,25,22)(23,28,27,26)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535162);