Finite Group Databasebeta

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#72232829

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8534
commutatorC22×C22⋊C4C22×C4
Frattini, cosocleC22×#121C23
prime radical#4122222C2
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G
LECSG ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1

4935 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 4935 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 16
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,4,8)(3,5,7,6)(9,10)(11,15,16,13)(12,14);(1,3,5,7,6,2,4,8)(9,10)(11,13)(12,14)(15,16);(2,7)(3,8)(4,5)(10,13)(14,15)

Polycyclic presentation:

2^10|a^2=d,b^2=df,e^2=i,b^a=bf,c^a=ce,c^b=ce,d^b=dg,d^c=dh,e^a=ehi,e^c=ei,e^d=eij,f^a=fg,f^b=fg,f^c=fhi,f^d=fi,f^e=fij,g^a=gi,g^b=gi,h^a=hij,h^b=hij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72232837, #72233660, #72235271, …
maximal normal subgroup of#72232837, #72233660, #72235271

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