Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13096 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1930 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)(5,15,20,44)(6,21,24,40)(7,25,16,12)(8,13)(9,14)(10,18)(11,23,36,30)(17,46)(19,43,29,60)(22,55)(26,34)(27,54)(28,57,45,38)(31,41)(32,61,64,66)(33,51)(35,56)(37,59)(39,58)(42,52)(47,53,50,63)(48,62)(49,65);(3,5,16,18,9,20,7,26)(4,10,25,44,14,34,12,15)(6,17,22,50,24,49,27,47)(8,29,45,58,31,19,28,51)(11,32,35,62,36,64,37,42)(13,39,57,43,41,33,38,60)(21,53,54,46,40,63,55,65)(23,52,56,66,30,48,59,61);(3,6)(4,11)(5,17)(7,27)(8,30)(9,24)(10,32)(12,37)(13,40)(14,36)(15,42)(16,22)(18,50)(19,52)(20,49)(21,41)(23,31)(25,35)(26,47)(28,56)(29,48)(33,53)(34,64)(38,54)(39,63)(43,65)(44,62)(45,59)(46,60)(51,66)(55,57)(58,61)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401940);