Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C25 |
| commutator | C24 | C24×C4 |
| prime radical | C23×D42 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
61697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(14,17)(15,19)(21,22)(23,26)(24,25)(27,28);(6,8)(10,11)(15,19)(18,20)(22,24)(26,28);(1,2,3,4)(5,6,7,8)(9,10,12,11)(14,17)(15,19)(21,23)(22,26)(24,28)(25,27);(2,4)(6,8)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)(13,15,16,19)(14,18,17,20)(21,24,25,22)(23,26,27,28)Polycyclic presentation:
2^10|a^2=g,c^2=i,e^2=fgj,b^a=bgh,c^a=cgi,c^b=cf,d^a=dhij,d^c=dhij,e^a=efj,e^b=efj,e^c=efgij,e^d=ehi160 conjugacy classes — view the full character table.
| maximal subgroup of | #72334421, #72334422, #72334425, … |
| maximal normal subgroup of | #72334421, #72334422, #72334425, … |
| maximal quotient of | C2×#72334401, #72337400, #72337401 |
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.7);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);