Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31419 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1880 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)(7,24)(8,28)(9,14)(10,18)(11,22)(12,26)(13,32)(16,44)(17,48)(19,35)(21,37)(23,42)(25,58)(27,43)(29,41)(30,38)(31,40)(33,50)(34,49)(36,47)(39,51)(45,59)(46,52)(53,55)(54,60)(56,62)(57,61)(63,65)(64,66);(3,5,8,29,9,18,32,19)(4,10,13,41,14,15,28,35)(6,16,22,50,23,46,30,47)(7,25,26,49,27,17,31,51)(11,36,20,52,38,33,42,44)(12,39,24,48,40,34,43,58)(21,54,55,64,56,45,57,65)(37,59,61,66,62,60,53,63);(3,6)(4,11)(5,16,18,46)(7,21)(8,30)(9,23)(10,33,15,36)(12,37)(13,42)(14,38)(17,45,25,54)(19,50,29,47)(20,28)(22,32)(24,53)(26,57)(27,56)(31,55)(34,59,39,60)(35,44,41,52)(40,62)(43,61)(48,66,58,63)(49,65,51,64);(3,7,9,27)(4,12,14,40)(5,17,18,25)(6,21,23,56)(8,31,32,26)(10,34,15,39)(11,37,38,62)(13,43,28,24)(16,45,46,54)(19,51,29,49)(20,53,42,61)(22,57,30,55)(33,59,36,60)(35,58,41,48)(44,63,52,66)(47,65,50,64)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411869);