Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,8,13,9,14,20,28,10,12,21,25,22,26,33,35)(7,15,17,27,18,29,31,36,16,11,19,23,30,24,32,34);(5,7)(6,11)(8,19)(9,18)(10,16)(12,15)(13,27)(14,24)(17,21)(20,32)(22,30)(23,25)(26,29)(28,36)(31,33)(34,35);(7,16)(11,15)(17,19)(18,30)(23,27)(24,29)(31,32)(34,36)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400137);