Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8535 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×#123 | C23 |
| prime radical | #4975876 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1 | |
1924 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 1924 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,8)(4,9)(5,11)(6,10)(7,14,21,30)(13,26)(16,27,28,24)(17,20)(18,31,32,22)(19,25)(23,29);(1,3,13,28)(2,7,11,22)(4,14,29,32)(5,16,6,15)(8,23,24,25)(9,12,20,27)(10,21,26,31)(17,18,19,30);(2,8)(4,17)(7,9)(10,12)(11,24)(14,18)(19,29)(20,22)(21,25)(23,31)(26,27)(30,32)Polycyclic presentation:
2^10|a^2=f,b^2=hij,d^2=j,b^a=bd,c^a=ce,c^b=cghi,d^c=dg,e^a=eh,e^b=egj,e^d=ei,f^b=fj,f^c=fh,f^e=fj,g^a=gi,h^a=hj58 conjugacy classes — view the full character table.
| maximal subgroup of | #72324313, #72324332, #72324342 |
| maximal normal subgroup of | #72324313, #72324332, #72324342 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);