Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×#13647 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | D4×C16 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,8,14,9,15,30,44,7,16,27,45,28,47,58,65,10,17,31,46,32,48,60,66,29,13,40,42,59,43,34,61)(5,18,20,49,21,24,38,55,22,36,51,62,53,26,39,57,23,41,52,64,54,56,37,63,19,11,33,35,50,6,12,25);(3,5)(4,11)(6,15)(7,22)(8,20)(9,21)(10,23)(12,34)(13,41)(14,35)(16,18)(17,36)(19,29)(24,47)(25,44)(26,48)(27,51)(28,53)(30,38)(31,52)(32,54)(33,40)(37,60)(39,58)(42,64)(43,56)(45,49)(46,62)(50,59)(55,65)(57,66)(61,63);(3,6,8,25,9,18,30,49,7,24,27,55,28,36,58,62,10,26,31,57,32,41,60,64,29,56,40,63,59,11,34,35)(4,12,14,5,15,20,44,21,16,38,45,22,47,51,65,53,17,39,46,23,48,52,66,54,13,37,42,19,43,33,61,50)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419943);