#72237933

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235864
commutator	#2308	C₂²×C₄
Frattini, cosocle	C₂×#2308	C₂³
prime radical	#72237871	C₂
LCS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#2308 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

3830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=de,d^2=i,f^2=ijk,b^a=bf,c^a=cd,c^b=cfg,d^b=dh,d^c=di,e^b=eij,e^c=ei,f^a=fk,f^b=fijk,f^c=fhij,f^d=fjk,g^a=gijk,g^c=gij,g^d=gk,h^a=hj,h^c=hj,h^d=hk,h^f=hk,i^b=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237948, #72237971, #72237998
maximal normal subgroup of	#72237948, #72237971, #72237998

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