Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #1304 |
| commutator | C22×C4 | C43 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | C4×#1304 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C23×#131 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2663 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 2663 conjugacy classes of subgroups — too many to draw. Only its 287 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)(3,8)(4,17)(5,21,23,10)(6,11)(7,12)(9,19)(13,34)(14,38,40,26)(15,27)(16,28)(18,44,46,43)(20,30)(22,42,49,32)(24,33)(25,36)(29,45,48,63)(31,47)(35,57,59,56)(37,51)(39,55,62,53)(41,54)(50,58,61,64)(52,60);(1,3,6,15,7,16,24,41)(2,8,11,27,12,28,33,54)(4,13,19,36,20,37,47,60)(5,14,22,39,23,40,49,62)(9,25,30,51,31,52,17,34)(10,26,32,53,21,38,42,55)(18,35,45,58,46,59,63,64)(29,50,44,57,48,61,43,56);(1,4,5,18)(2,9,10,29)(3,13,14,35)(6,19,22,45)(7,20,23,46)(8,25,26,50)(11,30,32,44)(12,31,21,48)(15,36,39,58)(16,37,40,59)(17,42,43,33)(24,47,49,63)(27,51,53,57)(28,52,38,61)(34,55,56,54)(41,60,62,64)Polycyclic presentation:
2^10|a^2=g,b^2=e,c^2=d,e^2=j,c^a=cef,d^a=dh,f^a=fi,f^c=fhj,g^c=gij,g^d=gj,h^a=hj,i^c=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72317672, #72317779 |
| maximal normal subgroup of | #72317672, #72317779 |
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.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);