#72339061

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#500057
commutator, Frattini, cosocle	#2297	C₂⁴
LCS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #35527 ◁ G
LECS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=g,b^2=g,c^2=ik,e^2=k,f^2=ik,h^2=k,c^a=cf,c^b=ce,d^a=dg,d^b=dijk,e^a=eh,e^b=ei,f^b=fhj,f^c=fik,f^d=fik,f^e=fj,g^c=gik,g^e=gk,h^c=hj,h^d=hk,h^e=hk,i^b=ik,j^b=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339071
maximal normal subgroup of	#72339071

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