Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13401 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2034 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1052 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)(5,15)(6,21)(7,26)(8,30)(9,14)(10,17)(11,24)(12,29)(13,28)(16,44)(18,34)(19,35)(20,36)(22,55)(23,57)(25,54)(27,40)(31,39)(32,53)(33,49)(37,60)(38,59)(41,58)(42,46)(43,50)(45,66)(47,52)(48,61)(51,65)(56,63)(62,64);(3,5)(4,10)(6,22)(7,19)(8,18)(9,20)(11,33)(12,36)(13,34)(14,35)(15,26)(16,25)(17,29)(21,52)(23,45)(24,56)(27,47)(28,50)(30,61)(31,62)(32,51)(37,42)(38,64)(39,63)(40,44)(41,65)(43,53)(46,59)(48,58)(49,60)(54,66)(55,57);(3,6,9,25)(4,11,14,38)(5,16)(7,27,29,23)(8,31,32,60)(10,33)(12,39,26,37)(13,40,41,57)(15,42)(17,47)(18,49)(19,45)(20,22)(21,53,54,30)(24,58,59,28)(34,44)(35,64)(36,63)(43,66)(46,50)(48,56)(51,62)(52,61)(55,65)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406232);