Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,10,11,9,7,8,12)(13,14,19,25,20,27,36,26)(15,22,30,42,31,39,44,43)(16,32,28,21,34,40,29,41)(17,35,37,23,18,38,33,24);(1,2,3,4)(5,7,10,12,9,6,8,11)(13,15,17,28,36,44,33,16,20,31,18,29,19,30,37,34)(14,21,24,39,25,40,35,43,27,41,23,22,26,32,38,42);(5,8,9,10)(6,11,7,12)(13,16,33,30,19,28,17,31,20,34,37,44,36,29,18,15)(14,22,38,41,26,43,23,40,27,39,35,21,25,42,24,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47963);