Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11468 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,16,22,17,23,29,36,18,24,14,20,15,21,30,37)(13,25,26,40,27,35,41,39,28,38,32,42,31,19,33,34);(3,5)(4,7)(6,10)(8,9)(11,13)(12,19)(14,26)(15,31)(16,32)(17,27)(18,28)(20,34)(21,38)(22,39)(23,25)(24,35)(29,33)(30,41)(36,42)(37,40);(11,14,17,30,18,16,15,29)(12,20,23,37,24,22,21,36)(13,26,27,41,28,32,31,33)(19,34,25,40,35,39,38,42)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387276);