Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C22×#3223 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,24,33)(16,26,28,36)(17,29,31,21)(20,30,35,27)(23,37,38,44)(25,41,40,34)(32,42,43,39);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,32)(21,34)(22,33)(26,37)(27,39)(28,38)(29,41)(30,42)(31,40)(35,43)(36,44);(7,12)(9,11)(13,16,18,28)(14,20,22,35)(15,23,24,38)(17,30,31,27)(19,32,33,43)(21,36,29,26)(25,42,40,39)(34,44,41,37);(6,10)(9,11)(15,24)(19,33)(23,38)(25,40)(32,43)(34,41)(37,44)(39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482693);