Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25 |
| direct product | C22×M128 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C32 | C24 |
| socle | C23 | C2×C32 |
| prime radical | C24 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)(34,66)(36,68)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420502);