Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93537 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1294 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C23×M16 | C22 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,15,16,23)(11,18)(12,13)(17,24,22,19)(20,21);(1,3)(2,5)(4,8)(6,7)(9,10)(11,19,20,24)(12,22)(13,17)(14,16)(15,21,23,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240722, #72240811, #72240812, … |
| maximal normal subgroup of | #72240722, #72240811, #72240812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 733);