Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #14657 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C26 | C24 |
| prime radical | C22×H(3,𝔽4) | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
3604 conjugacy classes of subgroups; only the 241 normal subgroups are shown.
This group has 3604 conjugacy classes of subgroups — too many to draw. Only its 241 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,6)(4,12)(5,9)(7,11)(8,13)(10,14)(15,16)(17,18)(19,23)(20,24)(21,25,33,41)(22,26)(27,34)(28,43,45,35)(29,36)(30,37)(31,40,46,38)(32,39)(42,47,44,48);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,19,22,29)(18,23,26,36)(20,30,32,27)(21,28,33,45)(24,37,39,34)(25,35,41,43)(31,44,46,42)(38,48,40,47);(1,3)(2,7)(4,10,13,16)(5,11)(6,9)(8,15,12,14)(17,20,22,32)(18,24,26,39)(19,27,29,30)(21,31)(23,34,36,37)(25,40)(28,44)(33,46)(35,47)(38,41)(42,45)(43,48);(1,4)(2,8,9,12)(3,10,11,16)(5,13)(6,14)(7,15)(17,21)(18,25,26,41)(19,28)(20,31)(22,33)(23,35,36,43)(24,38,39,40)(27,42)(29,45)(30,44)(32,46)(34,47,37,48)Polycyclic presentation:
2^10|a^2=g,b^2=ij,c^2=ghi,d^2=ef,b^a=bgh,c^a=chj,c^b=cghij,d^a=de,d^b=dhj,d^c=degi,e^a=ei,e^b=eij,e^c=eij,e^d=eij,f^a=fj,f^b=fi,f^c=fi,f^d=fij,g^d=gi,h^d=hj64 conjugacy classes — view the full character table.
| maximal subgroup of | #72249452, #72249467, #72249477, … |
| maximal normal subgroup of | #72249452, #72249467, #72249477, … |
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.9*F.10);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);