Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#12600 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | C22×C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,6)(7,8,10,9)(11,12)(13,17)(14,19)(15,20)(16,18)(21,28)(22,29)(23,27)(24,32)(25,30)(26,31)(33,39)(34,37)(35,38)(36,40)(41,42);(4,6)(11,13,14,21,15,22,24,33,16,23,25,34,26,35,36,41)(12,17,19,28,20,29,32,39,18,27,30,37,31,38,40,42);(7,9,10,8)(12,18)(17,27)(19,30)(20,31)(28,37)(29,38)(32,40)(39,42)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400128);