Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #20608 |
| commutator | C2×#41 | C23×C4 |
| Frattini, cosocle | C2×#131 | C24 |
| socle | C2 | #184912 |
| prime radical | #412 | C4 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,13)(8,22)(14,17)(15,27)(16,23)(24,29)(28,31)(30,32);(1,2)(3,14)(4,6)(5,10)(7,12)(8,16)(9,11)(13,17)(15,27)(18,21)(19,26)(20,25)(22,23)(24,29)(28,31)(30,32);(1,2)(4,6)(5,10)(7,12)(8,23)(9,11)(13,14)(18,21)(19,26)(20,25)(24,32)(27,28);(1,3,10,24,9,22,26,28,7,17,25,32,6,16,18,27)(2,8,20,31,11,14,21,30,12,23,5,15,4,13,19,29)Polycyclic presentation:
2^10|d^2=efij,f^2=i,g^2=j,i^2=j,c^a=chj,c^b=chj,d^a=de,d^b=dg,d^c=dei,e^b=eh,e^c=ej,f^b=fhj,g^a=ghj,g^b=gj,g^c=gh,g^e=gj,g^f=gj,h^d=hj106 conjugacy classes — view the full character table.
| maximal subgroup of | #72231730, #72281944, #72307752, … |
| maximal normal subgroup of | #72231730, #72281944, #72307752, … |
| maximal quotient of | #72239965, #72313995, C2×#72231607 |
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, 4, F.5*F.6*F.9*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);