Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24033 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C22×#576 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11,14,28)(8,20)(9,25)(12,26,21,15)(13,30,16,27)(18,35)(19,31)(22,24)(23,36)(29,33,32,34);(1,2)(3,4)(5,6,9,26,10,14,25,15)(7,16,19,33,17,13,31,34)(8,21,20,11,24,12,22,28)(18,32,35,27,23,29,36,30);(6,12)(7,17)(8,22)(9,25)(11,26)(13,32)(14,21)(15,28)(16,29)(18,35)(20,24)(23,36)(27,33)(30,34);(2,4)(5,7)(6,13)(8,23)(9,19)(10,17)(11,27)(12,29)(14,16)(15,33)(18,24)(20,36)(21,32)(22,35)(25,31)(26,34)(28,30)Polycyclic presentation:
2^10|a^2=g,b^2=gi,e^2=i,i^2=j,b^a=be,c^b=ceh,d^a=df,d^b=dfgj,d^c=dj,e^a=ei,e^b=ei,e^c=eij,e^d=ej,f^b=fj,g^b=gj,h^d=hj,i^a=ij,i^b=ij,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250436, #72252732, #72252755, … |
| maximal normal subgroup of | #72250436, #72252732, #72252755, … |
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, 2, F.7*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);