Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26505 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,7)(5,9,10,8)(11,12,18,24)(13,25,30,38)(14,31,34,41)(15,35,32,40)(16,23,17,22)(19,29,39,28)(20,27,42,33)(21,26,37,36);(4,7)(8,9)(11,13,17,29,18,30,16,28)(12,19,23,38,24,39,22,25)(14,32,33,36,34,15,27,26)(20,40,31,37,42,35,41,21);(11,14,17,33,18,34,16,27)(12,20,23,31,24,42,22,41)(13,26,29,32,30,36,28,15)(19,37,38,35,39,21,25,40);(3,5,6,10)(4,8,7,9)(11,15,17,26,18,32,16,36)(12,21,23,40,24,37,22,35)(13,27,29,14,30,33,28,34)(19,20,38,31,39,42,25,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351920);