Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93520 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C2×C4×C22⋊C4 | C22 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C22⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,13)(12,15)(14,16)(17,18,19,22)(20,23,24,21);(2,6,8,7)(10,13)(11,16)(14,15)(17,18)(19,22,24,21)(20,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=fg,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 789);