Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#20691 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4338 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)(5,8)(6,7)(9,10)(11,12)(13,17)(14,24,26,18)(15,19)(16,20)(21,35,37,29)(22,30)(23,31)(25,38,40,32)(27,33)(28,34)(36,42,39,41);(4,7)(8,10)(11,13,16,23)(12,17,20,31)(14,21,26,37)(15,27,28,22)(18,29,24,35)(19,33,34,30)(25,39,40,36)(32,42,38,41);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,21)(15,25)(16,26)(17,29)(19,32)(20,24)(22,36)(23,37)(27,39)(28,40)(30,41)(31,35)(33,42)(34,38);(11,15,16,28)(12,19,20,34)(13,22,23,27)(14,25,26,40)(17,30,31,33)(18,32,24,38)(21,36,37,39)(29,41,35,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278065);