Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 10556 conjugacy classes of subgroups; only the 3317 normal subgroups are shown.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(15,19)(18,20)(21,22)(23,24)(25,27)(26,28);(2,4)(5,6,9,12)(7,10,8,11)(14,17)(18,20);(5,7,9,8)(6,10,12,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,26)(27,28);(7,8)(10,11)(13,15,16,19)(14,18,17,20)(21,22)(23,28)(24,26)(25,27);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,24,25,28)(22,26,27,23);(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,27)(26,28)Polycyclic presentation:
2^10|b^2=h,c^2=hj,d^2=j,e^2=g,b^a=bh,c^b=cj,d^a=dj,d^c=dg,e^a=eghj,e^b=ei,e^d=ehj,f^b=fhj,f^c=fhj,f^e=fg124 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72278611 |
| maximal normal subgroup of | C2×#72278611 |
| maximal quotient of | #72278822, C2×#72278611, #72299068, … |
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.8);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);