Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102147 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
394 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 394 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) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,7,8,6)(10,13,14,15)(18,21)(19,22)(23,24);(2,5,7,6)(9,10,11,15,12,14,16,13);(17,18,22,24,20,23,19,21)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=gh,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9372);