Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 7783 conjugacy classes of subgroups; only the 1153 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(13,14)(15,18)(16,20)(17,19)(21,22,26,28)(23,24,27,25);(2,4)(6,8)(10,12)(13,15,17,16)(14,18,19,20)(21,23,26,27)(22,25,28,24);(2,4)(6,8)(9,10)(11,12)(14,19)(18,20)(21,24,26,25)(22,27,28,23);(1,2,3,4)(5,6,7,8)(13,16,17,15)(14,20,19,18)(21,25,26,24)(22,23,28,27);(1,2)(3,4)(5,6)(7,8)(13,14)(15,20)(16,18)(17,19)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=f,d^2=gi,b^a=bf,c^a=cfg,c^b=cj,d^a=dh,d^b=di,d^c=di,e^a=efh,e^b=egi,e^c=egi,e^d=egi148 conjugacy classes — view the full character table.
| maximal subgroup of | #72323723, #72323730, #72323750, … |
| maximal normal subgroup of | #72323723, #72323730, #72323750, … |
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.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);