Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #500177 |
| commutator, Frattini, cosocle | C2×#127 | C24 |
| prime radical | #7487867 | C2 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1570 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,8,15,29)(4,9)(5,10)(6,12)(7,11)(13,27,42,55)(14,44,45,35)(16,30,17,31)(18,36)(19,33)(20,34)(21,52,46,28)(22,32)(23,37)(24,39)(25,38)(26,40)(41,62,63,58)(43,56,47,57)(48,64,53,54)(49,60)(50,61)(51,59);(1,3)(2,8,11,29)(4,13)(5,21,23,14,24,46,25,45)(6,16)(7,15)(9,27,33,55)(10,35,37,52,39,44,38,28)(12,30,40,31)(17,26)(18,48,49,63,22,53,51,41)(19,42)(20,47)(32,58,59,54,36,62,60,64)(34,57,61,56)(43,50);(1,4)(2,9)(3,13)(5,22)(6,20)(7,19)(8,27)(10,36)(11,33)(12,34)(14,41)(15,42)(16,47)(17,43)(18,24)(21,53)(23,51)(25,49)(26,50)(28,54)(29,55)(30,57)(31,56)(32,39)(35,62)(37,60)(38,59)(40,61)(44,58)(45,63)(46,48)(52,64);(1,5)(2,10)(3,14,15,45)(4,18)(6,25)(7,24)(8,28,29,52)(9,32)(11,39)(12,37)(13,41,42,63)(16,46,17,21)(19,22)(20,51)(23,26)(27,54,55,64)(30,44,31,35)(33,36)(34,59)(38,40)(43,53,47,48)(49,50)(56,62,57,58)(60,61)Polycyclic presentation:
2^10|a^2=e,b^2=g,d^2=e,g^2=i,h^2=j,b^a=begi,c^a=ci,d^a=df,d^b=dfi,d^c=dej,e^b=ej,f^b=fh,f^c=fj,g^a=gij,g^d=ghj,g^f=gi,h^a=hi,h^b=hij,h^d=hj,h^f=hj,i^d=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72306877, #72306886, #72317834, … |
| maximal normal subgroup of | #72306877, #72306886, #72317834, … |
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.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);