Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 122 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)(5,6,8,11,9,7,10,12)(13,14,18,23,19,25,34,24)(15,26,28,43,29,21,40,42)(16,30,27,20,32,38,44,39)(17,33,35,22,36,41,31,37);(1,2,3,4)(5,7,8,12,9,6,10,11)(13,15,17,27,34,40,31,16,19,29,36,44,18,28,35,32)(14,20,37,21,23,38,33,42,25,39,22,26,24,30,41,43);(13,16,31,28,18,27,17,29,19,32,35,40,34,44,36,15)(14,21,41,20,24,43,22,30,25,26,33,39,23,42,37,38)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=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47923);