Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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);(3,4,10,16)(5,17,22,35)(6,23,27,42)(7,28,31,40)(8,15,9,14)(11,21,36,20)(12,32,39,26)(13,30,41,29)(18,37,24,33)(19,47,50,34)(25,55,57,38)(43,51,52,46)(44,49,60,48)(45,64,54,63)(53,58,62,56)(59,65,61,66);(3,5,9,21,10,22,8,20)(4,11,15,35,16,36,14,17)(6,24,26,51,27,18,32,46)(7,19,30,49,31,50,29,48)(12,37,23,52,39,33,42,43)(13,34,40,60,41,47,28,44)(25,54,56,66,57,45,58,65)(38,61,53,64,55,59,62,63);(3,6)(4,12)(5,18)(7,25)(8,26)(9,32)(10,27)(11,33)(13,38)(14,23)(15,42)(16,39)(17,43)(19,45)(20,46)(21,51)(22,24)(28,53)(29,56)(30,58)(31,57)(34,59)(35,52)(36,37)(40,62)(41,55)(44,63)(47,61)(48,65)(49,66)(50,54)(60,64);(3,7,9,30,10,31,8,29)(4,13,15,40,16,41,14,28)(5,19,21,49,22,50,20,48)(6,25,32,58,27,57,26,56)(11,34,35,60,36,47,17,44)(12,38,42,62,39,55,23,53)(18,45,51,66,24,54,46,65)(33,59,52,64,37,61,43,63)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354246);