Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133877 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,3,6,4,5,7,8)(9,10)(11,17,18,23)(12,20,21,15)(13,16)(14,19)(22,24);(2,5)(3,7)(11,18)(14,22)(17,23)(19,24);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24)Polycyclic presentation:
2^9|a^2=f,c^2=h,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233907, #72234524 |
| maximal normal subgroup of | #72233907, #72234524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41275);