Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27807 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2032 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)(11,12,15,18,17,21,16,20)(13,22,23,19,25,14,24,26);(3,5,6,9)(4,7,8,10)(12,18,21,20)(13,23,25,24);(4,8)(7,10)(11,13)(12,19)(14,20)(15,24)(16,23)(17,25)(18,22)(21,26);(11,14,15,26,17,22,16,19)(12,13,20,24,21,25,18,23)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364896);