Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C16 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13549 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C22 |
| commutator | C2 | C22×C4×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C23 | C22×C16 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 70 points:
(1,2);(7,8,10,15,11,16,23,33,12,17,24,34,26,36,45,55,13,18,25,35,27,37,46,56,28,38,47,57,48,58,63,68)(9,14,20,30,21,31,42,52,22,32,43,53,44,54,62,67,19,29,39,49,40,50,59,64,41,51,60,65,61,66,69,70);(3,4,5,6)(7,9)(8,14)(10,20)(11,21)(12,22)(13,19)(15,30)(16,31)(17,32)(18,29)(23,42)(24,43)(25,39)(26,44)(27,40)(28,41)(33,52)(34,53)(35,49)(36,54)(37,50)(38,51)(45,62)(46,59)(47,60)(48,61)(55,67)(56,64)(57,65)(58,66)(63,69)(68,70);(9,19)(14,29)(20,39)(21,40)(22,41)(30,49)(31,50)(32,51)(42,59)(43,60)(44,61)(52,64)(53,65)(54,66)(62,69)(67,70)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419738);