Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #111458 |
| prime radical | #6437628 | C2 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
4291 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 4291 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(2,7)(8,20,23,32)(9,24)(10,22)(11,26,27,31)(12,15)(21,25)(28,29);(1,2)(3,11,14,27)(4,7)(5,9)(6,10)(8,13,23,30)(12,26,15,31)(16,21)(17,22)(18,24)(19,25)(20,29,32,28);(1,3)(2,8,10,23)(4,15,17,12)(5,13)(6,14)(7,20)(9,11,25,27)(16,29,18,28)(19,30)(21,31)(22,32)(24,26)Polycyclic presentation:
2^10|a^2=e,b^2=h,c^2=gi,f^2=j,b^a=bd,c^a=ceg,c^b=cf,d^a=dh,d^c=dhi,e^b=eh,e^c=ei,f^a=fi,f^d=fj,g^a=gi,g^d=gj,h^c=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72238006, #72240507, C2×#72238003, … |
| maximal normal subgroup of | #72238006, #72240507, C2×#72238003, … |
| maximal quotient of | C2×#72238003 |
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.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);