Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #106766 |
| commutator | C22×D4 | C2×C42 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | C2×#3498 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
3262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15)(5,19,20,9)(6,10)(7,13,22,30)(8,18)(12,25)(14,26)(16,23,17,24)(21,28,31,27)(29,32);(3,12,14,29)(7,21,22,31)(11,25,26,32)(13,28,30,27);(1,3)(2,7)(4,12,18,29)(5,13)(6,14)(8,21,15,31)(9,11)(10,22)(16,27,17,28)(19,26)(20,30)(23,32,24,25)Polycyclic presentation:
2^10|a^2=g,b^2=hj,c^2=ef,d^2=j,c^a=ce,c^b=cd,d^a=dh,d^c=dh,e^a=ei,e^b=ehj,e^d=ej,f^a=fi,g^c=gi,g^e=gj,g^f=gj,h^c=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72238017, #72241317, #72241340, … |
| maximal normal subgroup of | #72238017, #72241317, #72241340, … |
| maximal quotient of | C2×#72238010 |
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.7);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, 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, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);