Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142540 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #18164 | C2 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 24 points:
(2,5)(3,7)(6,8)(15,20)(17,24);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(3,7)(5,8)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17)Polycyclic presentation:
2^9|c^2=i,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231776, #72262575, #72262739, … |
| maximal normal subgroup of | #72231776, #72262575, #72262739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49737);