Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,19,16,20,29,36)(13,17,22,31,23,32,40,42)(14,24,26,34,27,35,28,18)(21,37,38,33,25,41,39,30);(4,7)(8,9)(11,13,15,22,16,23,29,40)(12,17,19,31,20,32,36,42)(14,25,26,39,27,21,28,38)(18,33,24,41,34,30,35,37);(11,14,16,27)(12,18,20,34)(13,21,23,25)(15,28,29,26)(17,30,32,33)(19,35,36,24)(22,39,40,38)(31,41,42,37);(3,5,6,10)(4,8,7,9)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355265);