Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#22457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4)(5,15)(6,20,23,11)(7,12)(8,27)(9,14)(10,18)(13,30)(16,43,46,41)(17,42)(19,39)(21,50,54,34)(22,35,57,51)(24,58)(25,40)(26,38)(28,33)(29,37)(31,45,49,55)(32,48)(36,47)(44,60,64,63)(52,59,65,66)(53,61,56,62);(3,5,9,19)(4,10,14,33)(6,16,23,46)(7,24,26,48)(8,28,30,18)(11,31,20,49)(12,36,38,42)(13,39,27,15)(17,29,47,25)(21,52,54,65)(22,55,57,45)(32,40,58,37)(34,60,50,63)(35,43,51,41)(44,56,64,53)(59,62,66,61);(3,6)(4,11)(5,16)(7,21)(8,22)(9,23)(10,31)(12,34)(13,35)(14,20)(15,41)(17,44)(18,45)(19,46)(24,52)(25,53)(26,54)(27,51)(28,55)(29,56)(30,57)(32,59)(33,49)(36,60)(37,61)(38,50)(39,43)(40,62)(42,63)(47,64)(48,65)(58,66);(3,7,9,26)(4,12,14,38)(5,17,19,47)(6,21,23,54)(8,29,30,25)(10,32,33,58)(11,34,20,50)(13,40,27,37)(15,42,39,36)(16,44,46,64)(18,48,28,24)(22,56,57,53)(31,59,49,66)(35,62,51,61)(41,63,43,60)(45,65,55,52)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286379);