Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2539 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,8,10,7,5,6,9)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(3,5,7,4)(6,9,8,10)(11,13)(12,17,19,25)(14,22)(15,23)(16,21)(18,20,24,26);(6,8)(9,10)(13,21)(17,25)(20,26)(22,23)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107822);