Finite Group Databasebeta

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#72237356

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#134045
commutatorC26C22×C4
Frattini, cosocleC2×#379C23
prime radical#62817C4
LCSG ▷ C26 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2849 ◁ G
LECSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

2627 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 2627 conjugacy classes of subgroups — too many to draw. Only its 91 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=d,c^2=i,d^2=hij,b^a=be,c^a=ci,c^b=cf,d^a=dg,d^c=dh,e^b=eg,e^c=ei,e^d=ei,f^a=fij,f^b=fh,f^c=fj,f^d=fj,g^b=gi,g^c=gj,g^d=gj,h^a=hj,h^b=hj,i^b=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237358, #72237361, #72237385
maximal normal subgroup of#72237358, #72237361, #72237385

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