Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #8541 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| socle | C2 | C2×#8541 |
| prime radical | #10574874 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
12105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9)(3,7,13,21)(4,8,16,23)(6,10,19,25)(11,20,26,30)(12,18,27,14)(15,22,28,31)(17,24,29,32);(1,3)(2,7)(4,15)(5,18)(6,14)(8,22)(9,21)(10,12)(11,17)(13,19)(16,26)(20,24)(23,31)(25,27)(28,29)(30,32);(1,4,6,17)(2,8,10,24)(3,11,14,15)(5,16,19,29)(7,20,12,22)(9,23,25,32)(13,26,18,28)(21,30,27,31);(1,2)(3,12,14,7)(4,8)(5,9)(6,10)(11,22,15,20)(13,21,18,27)(16,23)(17,24)(19,25)(26,30,28,31)(29,32)Polycyclic presentation:
2^10|a^2=e,c^2=j,d^2=i,b^a=bg,c^b=cj,d^a=def,d^b=dhj,e^b=eh,e^d=ei,f^a=fi,f^b=fh,f^d=fi,g^a=gh,g^e=gj,h^a=hj,h^d=hj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72233466, #72258337, #72258396, … |
| maximal normal subgroup of | #72233466, #72258337, #72258396, … |
| maximal quotient of | C4×#133839, #72266614, C2×#72233465 |
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, 3, F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);