Finite Group Databasebeta

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#72321499

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72266274
commutator#2308C22×C4
Frattini, cosocleC2×#2308C23
prime radicalC24×D4C2×D4
LCSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C26 ◁ C2×#2308 ◁ G
LECSG ▷ C2×#2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

2510 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 2510 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,8)(4,7)(5,16,18,9)(6,10)(11,23,26,29)(12,19)(13,24)(14,20,17,28)(15,21)(22,25,32,30)(27,31);(2,7)(3,11)(8,22,24,32)(9,20)(10,21)(12,25)(13,26)(16,28)(19,23,31,29)(27,30);(1,3,6,13)(2,8,10,24)(4,12)(5,11,18,26)(7,19)(9,23,16,29)(14,25)(15,27)(17,30)(20,22)(21,31)(28,32)

Polycyclic presentation:

2^11|a^2=g,b^2=f,c^2=ghik,d^2=i,e^2=ij,b^a=bd,c^a=cg,c^b=ce,d^c=dh,e^a=ehijk,e^d=ejk,f^a=fi,f^c=fij,g^b=gi,g^e=gk,h^b=hj,h^c=hk,h^d=hk,h^e=hk,i^c=ik,j^a=jk,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321647, #72321780
maximal normal subgroup of#72321647, #72321780

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