Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#322 |
| commutator | C2×C8 | C22×C4×C8 |
| Frattini, cosocle | #1311 | C24 |
| socle | C22 | C22×#1313 |
| prime radical | C2×#41 | #322 |
| LCS | G ▷ C2×C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #479899 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 4 generators on 96 points:
(1,2,4,8,5,9,14,20,6,10,15,21,16,22,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32)(35,45,46,67,48,69,49,70)(36,50,51,71)(37,53)(40,56,58,81,57,80,60,83)(42,64,61,85)(43,62)(47,73,74,94,75,93,72,68)(52,78,77,76)(54,79)(59,87,88,95,84,96,86,82)(65,92,89,91)(66,90);(1,3)(2,7)(4,12)(5,13)(6,11)(8,18)(9,19)(10,17)(14,25)(15,23)(16,24)(20,29)(21,27)(22,28)(26,31)(30,32)(33,34)(35,40)(36,42)(37,43)(38,44)(39,41)(45,56)(46,58)(47,59)(48,57)(49,60)(50,64)(51,61)(52,65)(53,62)(54,66)(55,63)(67,81)(68,82)(69,80)(70,83)(71,85)(72,86)(73,87)(74,88)(75,84)(76,91)(77,89)(78,92)(79,90)(93,96)(94,95);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)(33,35,36,47,37,45,52,68,38,49,50,74,54,70,76,73,39,48,51,75,53,69,77,94,55,46,71,72,79,67,78,93)(34,40,42,59,43,56,65,82,44,60,64,88,66,83,91,87,41,57,61,84,62,80,89,95,63,58,85,86,90,81,92,96);(34,41)(40,57)(42,61)(43,62)(44,63)(56,80)(58,60)(59,84)(64,85)(65,89)(66,90)(81,83)(82,95)(86,88)(87,96)(91,92)Polycyclic presentation:
2^11|a^2=e,c^2=f,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^a=ch,c^b=ci,d^b=dk,e^c=ej,f^a=fj,f^e=fk,g^c=gk,h^a=hk260 conjugacy classes — view the full character table.
| maximal subgroup of | #72231918, #72231972, #72323144 |
| maximal normal subgroup of | #72231918, #72231972, #72323144 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);