Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#11229 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8)(9,18)(10,23,26,37)(11,27)(12,16)(13,17)(14,20)(15,25,30,24)(19,33,36,28)(21,29)(22,32)(31,35,38,34);(3,4)(5,6)(7,9,11,20,13,22,12,21)(8,14,16,18,17,29,27,32)(10,19,24,34,26,36,25,35)(15,28,23,31,30,33,37,38);(4,6)(7,10,11,24,13,26,12,25)(8,15,16,23,17,30,27,37)(9,19,20,34,22,36,21,35)(14,28,18,31,29,33,32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265226);