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 | C2×#32898 |
| commutator, Frattini, cosocle | C22×D4 | C25 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
At least 6997 conjugacy classes of subgroups; only the 963 normal subgroups are shown.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(4,15)(5,18)(8,11)(13,28)(16,29)(21,25)(24,32);(1,2,6,7)(3,11,14,8)(4,9,15,19)(5,10,18,20)(12,21,27,25)(13,26,28,22)(16,23,29,31)(17,24,30,32);(5,18)(10,20)(12,27)(13,28)(16,29)(21,25)(22,26)(23,31);(1,3)(2,8)(4,16)(5,13)(6,14)(7,11)(9,23)(10,22)(12,17)(15,29)(18,28)(19,31)(20,26)(21,24)(25,32)(27,30);(1,4)(2,9)(3,12)(5,17)(6,15)(7,19)(8,21)(10,24)(11,25)(13,16)(14,27)(18,30)(20,32)(22,23)(26,31)(28,29)Polycyclic presentation:
2^10|b^2=j,b^a=bj,d^b=di,d^c=dij,e^a=eh,e^b=ehj,e^c=eg,e^d=ef,f^a=fj,f^c=fi,g^d=gi,g^f=gj,h^d=hj,i^e=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72238724, #72249661, #72251132, … |
| maximal normal subgroup of | #72238724, #72249661, #72251132, … |
| maximal quotient of | #72233207, #72234429, #72235490, … |
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.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*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);;
gap> SetConjugate(coll, 7, 6, F.7*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);