Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| prime radical | #7623725 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ #20495 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(12,21,25,26)(13,27)(17,31,23,29)(18,34)(22,28)(24,36,38,37)(30,35)(32,40,33,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,23)(13,18)(14,16)(15,19)(17,25)(21,29)(22,35)(24,33)(26,31)(27,34)(28,30)(32,38)(36,39)(37,40);(11,14)(12,24,25,38)(13,28)(16,20)(17,32,23,33)(18,30)(21,36,26,37)(22,27)(29,39,31,40)(34,35);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,24)(13,27)(14,15)(17,33)(19,20)(21,36)(23,32)(25,38)(26,37)(29,40)(30,35)(31,39);(2,5)(6,8)(9,12,13,24,14,26,28,36,15,25,27,38,11,21,22,37)(10,17,18,32,16,29,30,40,19,23,34,33,20,31,35,39)Polycyclic presentation:
2^10|a^2=f,c^2=f,d^2=h,e^2=g,g^2=i,i^2=j,b^a=bh,d^a=dh,d^b=dj,d^c=dj,e^a=eij,e^b=ehj,e^c=eg,e^d=egi,f^e=fj,g^a=gj,g^c=gi,g^d=gij,i^c=ij,i^d=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72314251, #72314408, #72314497, … |
| maximal normal subgroup of | #72314251, #72314408, #72314497, … |
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.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);