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×#33739 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#2215 | 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)(14,22)(15,25)(17,27)(19,28)(20,32)(23,37)(24,35)(26,36)(29,38)(30,42)(31,41)(33,34)(39,40);(4,7)(8,10)(11,12)(13,17)(14,19)(15,20)(16,18)(21,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,39)(34,40)(35,38)(36,41)(37,42);(11,13,22,34,15,23,26,35,16,24,36,37,25,33,14,21)(12,17,28,40,20,30,31,38,18,29,41,42,32,39,19,27);(3,5,6,9)(4,8,7,10)(12,18)(17,29)(19,31)(20,32)(27,38)(28,41)(30,39)(40,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530428);