Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
745 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)(5,16,19,45)(6,12)(7,27,29,43)(8,14)(9,32)(10,15)(11,18)(13,30,41,31)(17,44,47,62)(20,40,21,51)(22,52,48,66)(23,54,55,64)(24,38)(25,58)(26,39)(28,42)(33,46)(34,49)(35,50)(36,53)(37,56,63,57)(59,61)(60,65);(3,5,10,19)(4,11,15,35)(6,22,26,48)(7,28,29,49)(8,21,9,20)(12,36,39,61)(13,40,41,51)(14,42,32,34)(16,43,45,27)(17,25,47,24)(18,30,50,31)(23,46,55,65)(33,58,60,38)(37,62,63,44)(52,64,66,54)(53,56,59,57);(3,6,9,25,10,26,8,24)(4,12,32,58,15,39,14,38)(5,17,21,48,19,47,20,22)(7,23,31,57,29,55,30,56)(11,33,42,61,35,60,34,36)(13,37,43,64,41,63,27,54)(16,44,51,66,45,62,40,52)(18,46,28,59,50,65,49,53)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388922);