Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1912 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);(3,4,10,16)(5,17,22,52)(6,23,27,48)(7,28,33,46)(8,15,9,14)(11,36,40,20)(12,34,43,26)(13,31,47,32)(18,38,24,61)(19,37,58,41)(21,59,35,44)(25,62,63,42)(29,49,53,39)(30,45,64,66)(50,56,65,55)(51,57,60,54);(3,5)(4,11)(6,24)(7,29)(8,21)(9,35)(10,22)(12,41)(13,44)(14,39)(15,49)(16,40)(17,28)(18,27)(19,25)(20,32)(23,60)(26,56)(30,57)(31,36)(33,53)(34,55)(37,43)(38,42)(45,65)(46,52)(47,59)(48,51)(50,66)(54,64)(58,63)(61,62);(3,6)(4,12)(5,18)(7,25)(8,34)(9,26)(10,27)(11,37)(13,42)(14,48)(15,23)(16,43)(17,50)(19,53)(20,54)(21,56)(22,24)(28,45)(29,58)(30,32)(31,64)(33,63)(35,55)(36,57)(38,59)(39,60)(40,41)(44,61)(46,66)(47,62)(49,51)(52,65);(3,6)(4,12)(5,19)(7,30)(8,26)(9,34)(10,27)(11,38)(13,45)(14,23)(15,48)(16,43)(17,51)(18,53)(20,55)(21,57)(22,58)(24,29)(25,32)(28,42)(31,63)(33,64)(35,54)(36,56)(37,59)(39,65)(40,61)(41,44)(46,62)(47,66)(49,50)(52,60)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530853);