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 | #9497 |
| commutator | C24×C4 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #115133 |
| prime radical | #6437616 | C2 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(8,22)(9,20)(10,21)(11,26)(19,24)(25,32)(27,29);(1,2,6,10)(3,11)(4,7,17,21)(5,9,18,25)(8,23)(12,26)(13,24)(14,27)(15,29)(16,20,31,32)(19,28)(22,30);(1,3)(2,8,10,24)(4,15,17,12)(5,13)(6,14)(7,19)(9,11,25,27)(16,30,31,28)(18,23)(20,29)(21,22)(26,32)Polycyclic presentation:
2^10|b^2=hj,c^2=g,f^2=j,b^a=bd,c^a=ce,c^b=cf,d^c=dh,e^b=eh,e^c=ei,g^a=gi,g^b=gj,g^d=gj,h^c=hj,i^b=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72238006, #72240535, C2×#72238005, … |
| maximal normal subgroup of | #72238006, #72240535, C2×#72238005, … |
| maximal quotient of | C2×#72238005 |
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, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 6, 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.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
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);