Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4472 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)(5,15,19,10)(6,20)(7,25,27,12)(8,13)(9,14)(11,23)(16,41,44,31)(17,32)(18,39,29,33)(21,48,47,34)(22,49,51,35)(24,40)(26,53,56,37)(28,38)(30,36)(42,45)(43,63)(46,58)(50,66)(52,62,55,60)(54,61)(57,64)(59,65);(3,5,8,19)(4,10,13,15)(6,21,23,47)(7,17,27,45)(9,29,28,18)(11,34,20,48)(12,32,25,42)(14,39,38,33)(16,30,44,24)(22,50,51,65)(26,54,56,46)(31,40,41,36)(35,59,49,66)(37,61,53,58)(43,55,64,52)(57,62,63,60);(3,6)(4,11)(5,16)(7,22)(8,23)(9,30)(10,31)(12,35)(13,20)(14,40)(15,41)(17,43)(18,47)(19,44)(21,29)(24,28)(25,49)(26,55)(27,51)(32,57)(33,48)(34,39)(36,38)(37,62)(42,63)(45,64)(46,65)(50,54)(52,56)(53,60)(58,66)(59,61);(3,7)(4,12)(5,17)(6,22)(8,27)(9,26)(10,32)(11,35)(13,25)(14,37)(15,42)(16,43)(18,46)(19,45)(20,49)(21,50)(23,51)(24,52)(28,56)(29,54)(30,55)(31,57)(33,58)(34,59)(36,60)(38,53)(39,61)(40,62)(41,63)(44,64)(47,65)(48,66)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345207);