Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1404 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)(5,8)(6,7)(9,10)(13,21)(14,25)(15,26)(17,28)(19,34)(20,33)(22,27)(23,38)(24,37)(29,32)(30,39)(31,42)(35,36)(40,41);(4,7)(8,10)(11,12,16,18)(13,17,24,30)(14,19,27,32)(15,20,26,33)(21,28,37,39)(22,29,25,34)(23,31,35,40)(36,41,38,42);(11,13,22,38,15,23,14,21,16,24,25,36,26,35,27,37)(12,17,29,42,20,31,19,28,18,30,34,41,33,40,32,39);(3,5,6,9)(4,8,7,10)(12,18)(17,30)(19,32)(20,33)(28,39)(29,34)(31,40)(41,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530432);