Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12927 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1852 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1022 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);(3,4,8,15,10,16,9,14)(5,17,20,44,21,45,32,39)(6,22,25,38,26,42,24,12)(7,27,29,41,31,40,30,13)(11,28,35,51,36,59,34,19)(18,46,49,64,50,62,48,43)(23,54,57,63,58,37,56,53)(33,47,52,55,61,66,60,65);(3,5,8,20,10,21,9,32)(4,11,15,35,16,36,14,34)(6,18,25,49,26,50,24,48)(7,28,29,51,31,59,30,19)(12,33,22,52,38,61,42,60)(13,39,27,17,41,44,40,45)(23,55,57,66,58,65,56,47)(37,62,53,43,54,46,63,64);(3,6)(4,12)(5,18)(7,23)(8,25)(9,24)(10,26)(11,33)(13,37)(14,42)(15,22)(16,38)(17,43)(19,47)(20,49)(21,50)(27,53)(28,55)(29,57)(30,56)(31,58)(32,48)(34,60)(35,52)(36,61)(39,62)(40,63)(41,54)(44,46)(45,64)(51,66)(59,65)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401454);