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 110 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 110 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)(13,14,19,25,20,27,37,26)(15,28,31,44,32,22,42,43)(16,33,30,41,35,40,29,21)(17,36,34,23,18,38,39,24);(1,2,3,4)(5,7,10,12,9,6,8,11)(13,15,17,29,37,42,39,35,20,32,18,30,19,31,34,16)(14,21,24,22,25,33,36,43,27,41,23,28,26,40,38,44);(5,8,9,10)(6,11,7,12)(13,16,34,31,19,30,18,32,20,35,39,42,37,29,17,15)(14,22,36,41,26,44,24,33,27,28,38,21,25,43,23,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=egi,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=chi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47967);