Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #133972 |
| commutator | C22×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#535 | C23 |
| LCS | G ▷ C22×#41 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1741 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1741 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,12)(3,13)(4,18,21,9)(5,23,25,10)(6,27,28,11)(8,29)(14,41)(15,45)(16,32)(17,33)(19,49)(20,35,53,48)(22,37,54,36)(24,42)(26,40,56,39)(30,57)(31,46)(34,51)(38,55)(43,59)(44,60)(47,63)(50,61)(52,62,64,58);(1,3)(2,8,11,13,12,32,27,33)(4,19,21,50)(5,24,25,15)(6,16)(7,17)(9,34,36,49,18,47,37,61)(10,38,39,42,23,31,40,45)(14,20)(22,51,54,63)(26,55,56,46)(28,29)(30,58,41,48,59,62,60,35)(43,52)(44,53)(57,64);(1,4)(2,9)(3,14,17,44)(5,20)(6,22)(7,21)(8,30,32,59)(10,35)(11,37)(12,18)(13,41,33,60)(15,19,24,50)(16,43,29,57)(23,48)(25,53)(26,52)(27,36)(28,54)(31,34,38,47)(39,62)(40,58)(42,61,45,49)(46,51,55,63)(56,64)Polycyclic presentation:
2^10|a^2=gj,b^2=e,c^2=g,d^2=j,e^2=i,h^2=j,b^a=bf,c^a=cgj,c^b=cd,d^a=dj,d^b=dg,e^a=eghj,e^c=egj,e^d=ej,f^a=fj,f^b=fgh,f^e=fi,g^b=gj,h^a=hj,h^b=hi,h^f=hj,i^a=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72319602, #72319679, #72319691, … |
| maximal normal subgroup of | #72319602, #72319679, #72319691, … |
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*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);