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#72324256

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72266325
commutatorC27C22×C4
Frattini, cosocleC2×#2846C23
prime radical#6499867C22
LCSG ▷ C27 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2846 ◁ G
LECSG ▷ C2×#2846 ▷ C25 ▷ C23 ▷ C2 ▷ C1

5184 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 5184 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=f,c^2=k,b^a=be,c^b=cfi,d^b=dg,e^a=eg,e^b=eh,e^c=eh,e^d=ei,f^a=fh,f^c=fk,f^d=fj,g^a=gi,g^b=gj,g^c=gj,h^a=hj,h^d=hk,i^b=ik,i^c=ik,j^a=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72324257, #72324260, #72324278
maximal normal subgroup of#72324257, #72324260, #72324278

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