Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#30554 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2773 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);(3,4,6,8)(5,9,10,7)(11,12)(13,15)(14,22)(16,18)(17,20)(19,25)(21,24)(23,26);(3,5,6,10)(4,7,8,9)(11,13,15,12)(14,19,23,21)(16,18,17,20)(22,24,26,25);(12,18)(13,20)(16,17)(19,26)(21,22)(24,25);(4,8)(5,10)(11,14)(12,19)(13,21)(15,23)(16,25)(17,24)(18,26)(20,22)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408328);