Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3144 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);(7,8,13,17,14,18,12,28)(9,19,21,15,10,22,24,31)(11,25,26,16,27,32,29,33)(20,30,36,37,23,38,35,34);(3,4)(5,6)(7,9,13,21,14,10,12,24)(8,15,17,22,18,31,28,19)(11,20,26,36,27,23,29,35)(16,30,32,37,33,38,25,34);(4,6)(7,10)(8,15)(9,14)(11,23)(12,24)(13,21)(16,30)(17,19)(18,31)(20,27)(22,28)(25,37)(26,36)(29,35)(32,34)(33,38);(7,11,14,27)(8,16,18,33)(9,20,10,23)(12,29,13,26)(15,30,31,38)(17,32,28,25)(19,34,22,37)(21,36,24,35)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346212);