Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #392135 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C2×H(5,𝔽2) | C24 |
| prime radical | #10586796 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×H(5,𝔽2) ▷ C22 ▷ C2 ▷ C1 | |
17510 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,25)(13,29)(15,24)(16,22)(17,20)(18,23)(19,26)(21,30)(27,28)(31,32);(1,3,4,13,6,12,16,28)(2,7,9,21,8,20,22,32)(5,17,19,30,18,14,15,31)(10,25,26,29,23,11,24,27);(2,8)(5,18)(7,20)(9,22)(11,25)(15,19)(21,32)(27,29);(3,12)(7,20)(11,25)(14,17)Polycyclic presentation:
2^10|b^2=e,e^2=j,b^a=bf,c^a=cj,c^b=cg,d^b=dh,e^a=ei,e^d=ei,f^b=fi,f^c=fij,f^e=fj,g^a=gi,g^f=gj,h^b=hi,h^e=hj,i^b=ij91 conjugacy classes — view the full character table.
| maximal subgroup of | #72226379, #72260582, #72308132, … |
| maximal normal subgroup of | #72226379, #72260582, #72308132, … |
| maximal quotient of | #72225836, #72226174, #72233157, … |
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, 2, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);