Finite Group Databasebeta

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#72323901

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227787
commutatorC22×#120C22×C4
Frattini, cosocleC2×#2613C23
prime radical#72266439C2
LCSG ▷ C22×#120 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G
LECSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1

3342 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 3342 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^11|a^2=d,b^a=be,c^a=chj,c^b=cf,d^b=dg,d^c=dik,e^a=eg,e^c=eh,e^d=ek,f^a=fh,f^d=fi,f^e=fj,g^a=gk,g^c=gi,g^f=gk,h^a=hi,h^b=hj,i^b=ik,j^a=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72323927, #72324330, #72324333
maximal normal subgroup of#72323927, #72324330, #72324333

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);