Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C29 · Schur multiplier (H2) = C235 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C4 | C28 |
| commutator, Frattini, cosocle | C2 | C29 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
8302839 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^10|i^2=j,b^a=bj,d^c=dj,f^e=fj,h^g=hj514 conjugacy classes — view the full character table.
| Fitting subgroup of | #72263258, #72265267, #72265383, … |
| maximal subgroup of | H(11,𝔽2), 2−1+10, #72263051, … |
| maximal normal subgroup of | H(11,𝔽2), 2−1+10, #72263051, … |
| maximal quotient of | C2×#72258201 |
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, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);