Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,8)(5,6)(9,10)(12,16)(14,17)(18,22)(19,23)(21,24)(25,26);(11,12,17,23,15,19,14,16)(13,18,24,26,20,25,21,22);(3,4)(5,8)(6,10)(7,9)(11,13)(12,18)(14,21)(15,20)(16,22)(17,24)(19,25)(23,26);(3,5,7,6)(4,8,9,10)(13,20)(18,25)(21,24)(22,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261220);