Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#25804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5010 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);(4,6)(7,8,13,19)(9,20,10,15)(11,26,28,33)(12,18,24,17)(14,23,30,22)(16,32,34,27)(21,36,25,29)(31,37,35,38);(7,9,12,22,13,10,24,23)(8,14,17,20,19,30,18,15)(11,21,27,37,28,25,32,38)(16,29,26,35,34,36,33,31);(3,4,5,6)(7,10)(8,15)(9,13)(11,25)(12,23)(14,17)(16,31)(18,30)(19,20)(21,28)(22,24)(26,29)(27,38)(32,37)(33,36)(34,35);(7,11,24,32,13,28,12,27)(8,16,18,33,19,34,17,26)(9,21,23,38,10,25,22,37)(14,29,15,31,30,36,20,35)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350472);