Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C230 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C26 | C26 |
| commutator | C25 | C25×C4 |
| prime radical | C211 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C1 | |
4467484379 conjugacy classes of subgroups; only the 533115 normal subgroups are shown.
This group has 4467484379 conjugacy classes of subgroups — too many to draw. Only its 533115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(5,6,7,8)(10,12)(14,16)(18,20)(22,24);(9,10)(11,12);(13,14)(15,16);(17,18)(19,20);(21,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^12|a^2=g,b^a=bh,c^a=ck,d^a=dj,e^a=egl,f^a=fi1120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72234537 |
| maximal normal subgroup of | #72234537 |
| automorphism group of | #60297 |
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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);