Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152017 |
| commutator | C2×C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #3152 | C23 |
| prime radical | #6350247 | C2 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C4×#41 ◁ #60818 ◁ G | |
| LECS | G ▷ #3152 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1528 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13)(4,15)(5,6)(7,11)(8,12)(9,17)(10,22)(14,27)(16,28)(18,24)(19,20)(21,26)(23,25)(29,30)(31,32);(1,2)(3,14,17,28,18,31,23,30)(4,19)(5,7)(6,11)(8,12)(9,16,24,32,25,29,13,27)(10,21)(15,20)(22,26);(1,3,15,31,6,23,19,28,8,18,10,14,7,17,26,30)(2,9,22,29,5,13,21,32,12,25,4,16,11,24,20,27)Polycyclic presentation:
2^10|b^2=gh,c^2=de,d^2=j,f^2=j,g^2=i,h^2=j,c^a=cf,c^b=cd,d^a=di,d^b=dhj,d^c=dg,e^b=eghi,e^c=eg,e^d=ehij,f^a=fj,f^b=fij,f^c=fij,f^d=fj,g^b=gi,g^c=gh,g^d=gi,g^e=gi,h^b=hi,h^c=hj,h^d=hj,h^e=hj,i^c=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72339850 |
| maximal normal subgroup of | #72339850 |
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.7*F.8);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);