Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #115204 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | #6470234 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
3388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(3,11,14,16)(4,15)(7,21,22,24)(8,23)(9,19)(12,27,31,30)(13,29,32,28)(17,18);(1,2)(3,12)(4,8)(5,19,20,9)(6,10)(7,13,22,32)(11,27)(14,31)(15,23)(16,30)(17,26,18,25)(21,28,24,29);(1,3)(2,7)(4,16,15,11)(5,13)(6,14)(8,24,23,21)(9,12)(10,22)(17,29,18,28)(19,31)(20,32)(25,30,26,27)Polycyclic presentation:
2^10|a^2=h,b^2=g,c^2=ef,b^a=bg,c^a=cd,c^b=ce,d^a=dj,d^b=dh,d^c=dh,e^a=ehi,e^b=ei,e^d=ej,f^a=fi,f^b=fi,g^c=gi,g^e=gj,g^f=gj,h^c=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72238017, #72241334, #72241339, … |
| maximal normal subgroup of | #72238017, #72241334, #72241339, … |
| maximal quotient of | C2×#72238015 |
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.8);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);