Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11357 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(7,8,9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,24)(22,25)(23,26);(3,4)(5,6)(11,13,15,14)(12,16,17,18)(20,24)(21,22)(25,26);(4,6)(11,14,15,13)(12,16,17,18)(20,25)(21,22)(24,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265754);