Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129837 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2274 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(2,5,6,7)(9,10,12,16,13,14,20,21)(11,17,19,24,18,15,22,23);(1,2)(3,7)(4,6)(5,8)(10,14)(15,17);(9,11)(10,15,14,17)(12,19)(13,18)(16,23,21,24)(20,22)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37035);