Finite Group Databasebeta

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#72291979

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#1343
commutatorC22×C22⋊C4C22×C4
Frattini, cosocleC2×#379C23
prime radical#394167C2
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C26 ◁ C22×#379 ◁ G
LECSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

5959 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 5959 conjugacy classes of subgroups — too many to draw. Only its 119 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)(7,14,16,10,17,19,15,20)(9,13,18,11);(2,4);(1,2,3,4)(5,7)(6,10,13,20)(8,15,18,16)(9,17)(11,14)(12,19)

Polycyclic presentation:

2^10|a^2=f,c^2=dgij,e^2=hij,f^2=hj,c^a=ce,c^b=cd,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

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72291984, #72292059, #72292064
maximal normal subgroup of#72291984, #72292059, #72292064

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.9*F.10);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetPower(coll, 6, 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, 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);