Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #14679 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C26 | C24 |
| prime radical | #10586796 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
17831 conjugacy classes of subgroups; only the 691 normal subgroups are shown.
This group has 17831 conjugacy classes of subgroups — too many to draw. Only its 691 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 — 4 generators on 64 points:
(1,2)(3,8)(4,9)(5,10)(6,13)(7,11)(12,26)(14,27)(15,28)(16,30)(17,31)(18,46)(19,33)(20,34)(21,35)(22,36)(23,52)(24,40)(25,38)(29,45)(32,42)(37,54)(39,55)(41,56)(43,57)(44,58)(47,60)(48,64)(49,62)(50,63)(51,61)(53,59);(1,3)(2,8)(4,18,20,14)(5,23,24,15)(6,16)(7,17)(9,32,34,57)(10,37,40,58)(11,31)(12,29)(13,30)(19,48,49,41)(21,42,22,43)(25,54,55,44)(26,45)(27,35,46,36)(28,38,52,39)(33,60,62,59)(47,51,53,50)(56,63,64,61);(1,4)(2,9)(3,14)(5,19)(6,22)(7,20)(8,27)(10,33)(11,34)(12,35)(13,36)(15,41)(16,42)(17,18)(21,26)(23,53)(24,50)(25,51)(28,56)(29,57)(30,32)(31,46)(37,64)(38,61)(39,62)(40,63)(43,45)(44,47)(48,54)(49,55)(52,59)(58,60);(1,5)(2,10)(3,15)(4,19)(6,24)(7,25)(8,28)(9,33)(11,38)(12,39)(13,40)(14,41)(16,23)(17,44)(18,47)(20,51)(21,49)(22,50)(26,55)(27,56)(29,37)(30,52)(31,58)(32,59)(34,61)(35,62)(36,63)(42,53)(43,48)(45,54)(46,60)(57,64)Polycyclic presentation:
2^10|b^2=g,b^a=bh,c^b=cf,d^b=de,e^a=ej,e^b=ei,f^a=fij,f^b=fj,g^c=gj,g^d=gi,h^c=hij,h^d=hj91 conjugacy classes — view the full character table.
| maximal subgroup of | #72228235, #72228243, #72228246, … |
| maximal normal subgroup of | #72228235, #72228243, #72228246, … |
| maximal quotient of | #72309400, #72317029 |
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, 2, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);