Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,15,20,24,18,16,21,12,17,14,11,22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,23)(14,21)(16,20)(17,22)(19,24);(9,12,15,22,16,13,14,24)(10,17,20,23,21,19,11,18)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31381);