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×#2596 |
| 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 | |
7118 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);(5,6,9,8)(7,11,12,10)(13,14,17,18)(15,20,19,16)(23,27)(26,28);(7,12)(10,11)(14,18)(15,19)(21,22,25,24)(23,26,27,28);(5,7,9,12)(6,10,8,11)(13,15,17,19)(14,16,18,20)(21,23)(22,26)(24,28)(25,27);(5,8,9,6)(7,11,12,10)(13,16)(14,19)(15,18)(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, 7535215);