#72321975

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72238480
commutator, Frattini, cosocle	#2308	C₂⁴
prime radical	#72216464	C₂
LCS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ #35346 ◁ G
LECS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4822 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:

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

Polycyclic presentation:

2^11|a^2=i,e^2=i,f^2=ij,b^a=be,c^a=cg,c^b=cf,d^a=dghij,d^b=dj,d^c=dgijk,e^a=ei,e^b=ei,e^c=eh,e^d=eik,f^a=fhi,f^b=fij,f^c=fij,f^d=fijk,f^e=fj,g^a=gk,g^b=gij,g^e=gk,h^b=hj,h^e=hk,h^f=hk,i^c=ik,j^a=jk,j^c=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322074, #72337750, #72337775
maximal normal subgroup of	#72322074, #72337750, #72337775

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