Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(8,13)(10,21)(11,22)(14,23)(15,24)(16,28)(17,27)(19,34)(20,33)(25,26)(29,32)(30,35)(31,38)(36,37);(7,8,14,28,11,16,10,13,12,17,21,26,22,25,23,27)(9,15,29,38,20,31,19,24,18,30,34,37,33,36,32,35);(4,6)(7,9)(8,15)(10,19)(11,20)(12,18)(13,24)(14,29)(16,31)(17,30)(21,34)(22,33)(23,32)(25,36)(26,37)(27,35)(28,38);(9,18)(15,30)(19,32)(20,33)(24,35)(29,34)(31,36)(37,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530165);