Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C82 |
| Frattini, cosocle | C2×C4×C4⋊C4 | C22 |
| socle | C23 | #342 |
| prime radical | C24 | #95 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C4⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 410 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) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(17,18)(19,22,24,21)(20,23);(9,10,11,13,12,14,15,16)(18,21,23,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=h,b^a=bc,c^a=cf,e^a=ei,f^a=fi,d^b=dfi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 479);