Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #6628896 |
| commutator, Frattini, cosocle | C22×D4 | C25 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3484 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
At least 18040 conjugacy classes of subgroups; only the 947 normal subgroups are shown.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(4,15)(5,19)(9,24)(14,28)(17,29)(20,25)(21,26)(23,32);(1,2)(3,11)(4,16,15,8)(5,9)(6,10)(7,12)(13,25,27,20)(14,26)(17,31,29,22)(18,32,30,23)(19,24)(21,28);(3,12)(7,11)(14,28)(17,29)(18,30)(21,26)(22,31)(23,32);(1,3)(2,7)(4,17)(5,14)(6,12)(8,22)(9,21)(10,11)(13,18)(15,29)(16,31)(19,28)(20,23)(24,26)(25,32)(27,30);(1,4)(2,8)(3,13)(5,18)(6,15)(7,20)(9,23)(10,16)(11,25)(12,27)(14,17)(19,30)(21,22)(24,32)(26,31)(28,29)Polycyclic presentation:
2^10|b^2=i,b^a=bi,d^b=dij,d^c=dj,e^a=eh,e^b=ehj,e^c=eg,e^d=ef,f^a=fj,f^c=fi,g^d=gij,g^f=gj,h^b=hj,h^d=hj,i^e=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72249661, #72258051, #72260582, … |
| maximal normal subgroup of | #72249661, #72258051, #72260582 |
| maximal quotient of | #72233212, #72234423, #72256140, … |
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.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);