Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25054 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2088 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)(5,7)(6,8)(9,10)(11,12)(13,17)(14,21)(15,19)(16,20)(18,25)(22,24)(23,26);(3,5,6,9)(4,7,8,10)(11,13,15,21,16,23,25,22)(12,14,18,17,20,24,19,26);(4,8)(7,10)(11,13)(12,17)(14,19)(15,21)(16,23)(18,24)(20,26)(22,25);(11,14,15,17,16,24,25,26)(12,13,18,21,20,23,19,22)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351316);