Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21098 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,16,13,17,11,27)(9,18,22,29,20,32,21,14)(10,23,25,30,26,37,24,15)(19,33,36,38,34,28,35,31);(7,9)(8,14,17,29)(10,19)(11,21)(12,22)(13,20)(15,28,30,33)(16,18,27,32)(23,31,37,38)(24,35)(25,36)(26,34);(4,6)(7,10,11,24,13,26,12,25)(8,15,16,23,17,30,27,37)(9,19,21,35,20,34,22,36)(14,28,18,31,29,33,32,38);(9,20)(14,29)(18,32)(19,34)(21,22)(28,33)(31,38)(35,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278900);