Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C24 |
| prime radical | C27 | D4 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
8597 conjugacy classes of subgroups; only the 265 normal subgroups are shown.
This group has 8597 conjugacy classes of subgroups — too many to draw. Only its 265 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,14)(7,15,9,18)(8,13,10,19)(12,16,20,17)(21,22,25,29)(23,26,30,28)(24,32,33,35)(27,34,36,31);(5,7,11,9)(6,12,14,20)(8,17,10,16)(13,18,19,15)(23,30)(24,33)(27,36)(32,35);(2,4)(5,8,11,10)(6,13,14,19)(7,16,9,17)(12,15,20,18)(21,23,25,30)(22,26,29,28)(24,34,33,31)(27,32,36,35);(2,4)(5,9,11,7)(6,12,14,20)(8,16,10,17)(13,18,19,15)(21,24,25,33)(22,27,29,36)(23,31,30,34)(26,35,28,32)Polycyclic presentation:
2^10|a^2=h,b^2=hj,c^2=h,d^2=h,f^2=h,g^2=hj,b^a=bg,c^a=ce,c^b=chi,d^a=defh,d^b=dj,d^c=dh,f^a=fh,f^b=fhi,f^d=fh,g^a=ghj,g^b=ghj,g^c=gj,g^d=ghij,g^f=gj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72227602, #72227750 |
| maximal normal subgroup of | #72227750 |
| maximal quotient of | #72227598, #72227689 |
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.8);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);