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 | |
28681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,19,16)(21,22)(23,24)(25,28)(26,27);(7,11)(10,12)(13,15,17,19)(14,18,20,16)(21,23,25,27)(22,26,28,24);(2,4)(6,9)(7,11)(13,16,17,18)(14,19,20,15)(21,24,25,26)(22,27,28,23);(5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,20)(17,18)(21,24)(22,23)(25,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^10|a^2=g,b^2=gh,c^2=gh,b^a=bh,c^a=cj,d^a=dg,d^b=dg,d^c=dj,e^c=ei,f^b=fj,f^c=fhj,f^e=fg130 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72293948 |
| maximal normal subgroup of | C2×#72293948 |
| maximal quotient of | #72293988, C2×#72293948 |
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.7);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*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);