Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,23,25,42)(16,27,30,41)(18,32,29,22)(19,34,35,24)(20,37,39,31)(26,43,33,38)(28,40,44,36);(1,3,4,2)(6,9)(10,12);(13,15,17,25)(14,19,21,35)(16,28,30,44)(18,33,29,26)(20,38,39,43)(22,40,32,36)(23,41,42,27)(24,37,34,31);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^c=fh,h^c=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402161);