Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #47031 |
| commutator, Frattini, cosocle | C23×C4 | C25 |
| socle | C2 | #4320266 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,10,6,9,21,11)(3,12,15,7,14,24,31,23)(4,16,19,25,18,27,20,8)(13,28,30,32,29,22,17,26);(8,25)(13,29)(16,27)(17,30);(1,3,6,14)(2,7)(4,17)(5,15,21,31)(8,26,25,32)(9,23)(10,24)(11,12)(13,19)(16,28,27,22)(18,30)(20,29);(1,4)(2,8)(3,13)(5,20)(6,18)(7,22)(9,25)(10,27)(11,16)(12,26)(14,29)(15,17)(19,21)(23,28)(24,32)(30,31);(2,9)(3,14)(7,23)(8,25)(10,11)(12,24)(13,29)(15,31)(16,27)(17,30)(22,28)(26,32)Polycyclic presentation:
2^10|a^2=g,c^2=fgh,g^2=j,h^2=j,b^a=bf,c^a=ch,c^b=cf,d^a=dg,d^b=dghij,d^c=dhj,e^a=eij,e^c=egh,f^d=fj,g^d=gj,h^a=hj,h^d=hj,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72251866, #72251900, #72251931, … |
| maximal normal subgroup of | #72251866, #72251900, #72251931, … |
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.6*F.7*F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);