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#72334849

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#4131225
commutator, Frattini, cosocleC25C25
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

3897 conjugacy classes of subgroups; only the 511 normal subgroups are shown.

normal subgroups only

This group has 3897 conjugacy classes of subgroups — too many to draw. Only its 511 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7,13,19)(4,8,15,23)(5,9)(6,10)(11,20)(12,21,16,24)(14,28,17,22)(18,25)(26,32)(27,30)(29,31);(7,19)(14,17)(20,30)(21,24)(22,28)(26,29);(1,3)(2,7,10,19)(4,11)(5,16,18,12)(6,13)(8,20,23,30)(9,24)(14,29,17,26)(15,27)(21,25)(22,32)(28,31);(1,4)(2,8)(3,11)(5,17)(6,15)(7,20)(9,22)(10,23)(12,26)(13,27)(14,18)(16,29)(19,30)(21,32)(24,31)(25,28);(1,5)(2,9)(3,12)(4,14)(6,18)(7,21)(8,22)(10,25)(11,26)(13,16)(15,17)(19,24)(20,31)(23,28)(27,29)(30,32)

Polycyclic presentation:

2^10|a^2=g,c^2=f,b^a=bh,c^a=chi,c^b=ci,d^a=dij,d^b=dfi,d^c=dfi,e^a=egh,e^b=egh,e^c=eh,e^d=ehij,f^a=fj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72334852, #72334917, #72337418
maximal normal subgroup of#72334852, #72334917, #72337418

GAP

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, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);