Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10547 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #394 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,8,10)(5,6,9,7)(11,12)(13,20)(14,22)(15,23)(16,19)(17,21)(18,25)(24,26);(3,5)(4,6)(7,10)(8,9)(11,13)(12,17)(14,18)(15,24)(16,22)(19,26)(20,23)(21,25);(4,9)(5,10)(6,7)(11,14,15,17,16,20,25,26)(12,13,18,21,19,22,23,24)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146452);