Finite Group Databasebeta

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#72291982

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11753
commutatorC22×C22⋊C4C22×C4
Frattini, cosocleC2×#379C23
prime radicalC27D4
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C26 ◁ C22×#379 ◁ G
LECSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

5388 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 5388 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 20
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(5,6,8,12)(7,13,14,10,15,18,20,17)(9,11,16,19);(2,4)(10,17)(13,18);(1,2,3,4)(5,7)(6,10,11,17)(8,14,16,20)(9,15)(12,13)(18,19)

Polycyclic presentation:

2^10|a^2=f,c^2=dgh,e^2=hij,f^2=hj,b^a=bhj,c^a=ce,c^b=cd,d^a=dj,e^a=eg,e^c=eh,f^c=fghij,f^e=fi,g^a=gi,g^c=gj,g^e=gj,g^f=gj,h^c=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72291984, #72292077, #72292078
maximal normal subgroup of#72291984, #72292077, #72292078

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