Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C216 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #6342371 |
| commutator, Frattini, cosocle | C26 | C25 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
225155 conjugacy classes of subgroups; only the 2294 normal subgroups are shown.
This group has 225155 conjugacy classes of subgroups — too many to draw. Only its 2294 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 — 5 generators on 64 points:
(1,2)(3,8)(4,9)(5,10)(6,13)(7,11)(12,24)(14,25)(15,26)(16,28)(17,29)(18,31)(19,32)(20,30)(21,33)(22,36)(23,34)(27,43)(35,47)(37,51)(38,49)(39,48)(40,50)(41,53)(42,54)(44,55)(45,56)(46,57)(52,61)(58,63)(59,64)(60,62);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,25)(10,26)(11,29)(12,27)(13,28)(18,38)(19,37)(20,40)(21,39)(22,41)(23,42)(24,43)(30,50)(31,49)(32,51)(33,48)(34,54)(35,52)(36,53)(44,59)(45,60)(46,58)(47,61)(55,64)(56,62)(57,63);(9,30)(10,34)(14,37)(15,41)(25,48)(26,52)(31,56)(32,33)(35,36)(38,59)(39,40)(42,61)(49,63)(50,51)(53,54)(55,57)(58,60)(62,64);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,25)(10,31)(11,32)(12,30)(13,33)(15,38)(16,39)(17,37)(20,24)(22,45)(23,46)(26,49)(27,50)(28,48)(29,51)(34,57)(35,55)(36,56)(40,43)(41,60)(42,58)(44,47)(52,64)(53,62)(54,63)(59,61);(1,5)(2,10)(3,15)(4,18)(6,22)(7,23)(8,26)(9,31)(11,34)(12,35)(13,36)(14,38)(16,41)(17,42)(19,46)(20,44)(21,45)(24,47)(25,49)(27,52)(28,53)(29,54)(30,55)(32,57)(33,56)(37,58)(39,60)(40,59)(43,61)(48,62)(50,64)(51,63)Polycyclic presentation:
2^11|c^a=ci,c^b=ch,d^c=dg,e^c=ef,f^a=fk,f^b=fj,g^a=gjk,g^b=gk,h^d=hk,h^e=hj,i^d=ijk,i^e=ik158 conjugacy classes — view the full character table.
| maximal subgroup of | UT(4,𝔽4), #72241458, #72242397, … |
| maximal normal subgroup of | UT(4,𝔽4), #72242720, #72338147 |
| maximal quotient of | #72334288 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);