Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| prime radical | C2×#15257 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 7089 conjugacy classes of subgroups; only the 1167 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(10,12)(21,22)(23,24)(25,26)(27,28);(6,8)(9,10)(11,12)(13,14)(15,20)(16,19)(17,18)(21,23)(22,24)(25,27)(26,28);(5,6,7,8)(14,18)(15,19)(22,26)(23,27);(2,4)(13,15)(14,16)(17,19)(18,20)(21,22)(23,28)(24,27)(25,26);(1,2)(3,4)(13,16)(14,19)(15,18)(17,20)(21,24,25,28)(22,27,26,23)Polycyclic presentation:
2^10|c^2=hi,e^2=hj,b^a=bh,c^a=chj,c^b=cf,d^b=dfhi,d^c=dfhi,e^a=ehj,e^d=efg154 conjugacy classes — view the full character table.
| maximal subgroup of | #72331199, #72331218, #72331237, … |
| maximal normal subgroup of | #72331199, #72331218, #72331237, … |
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, 3, F.8*F.9);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);