#72237937

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235868
commutator	#381	C₂×C₄²
Frattini, cosocle	C₂×#2308	C₂³
prime radical	#6451874	C₂²
LCS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#2308 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

3324 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=efgik,c^2=deh,d^2=ijk,f^2=ik,b^a=bf,c^a=cd,c^b=cdfh,d^a=dk,d^b=dh,d^c=di,e^b=ei,e^c=eij,f^a=fk,f^b=fk,f^c=fhk,f^d=fjk,g^a=gik,g^b=gik,g^c=gik,g^d=gk,h^a=hj,h^b=hj,h^c=hj,h^d=hk,h^f=hk,i^c=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237948, #72237974, #72238656, …
maximal normal subgroup of	#72237948, #72237974, #72238656, …

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