Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1188 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,18,21,19)(12,16)(13,23)(14,17)(15,20,24,22);(2,5)(3,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23);(2,5)(3,7)(9,10)(11,19)(12,16)(13,23)(14,17)(15,20)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226731 |
| maximal normal subgroup of | #72226731 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18633);