#72337710

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#251827
commutator, Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#72210498	C₂
LCS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

4761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^11|a^2=hj,b^2=fgk,c^2=jk,e^2=j,g^2=k,b^a=bf,c^a=ce,c^b=cjk,d^b=dk,d^c=dfg,e^b=ei,e^c=ej,e^d=eh,f^a=fk,f^b=fhj,f^c=fij,f^e=fj,g^a=ghjk,g^b=ghj,g^c=gij,g^f=gk,h^b=hk,h^c=hj,i^a=ij,i^b=ij,i^d=ik

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338775, #72339010, #72340523
maximal normal subgroup of	#72338775, #72339010, #72340523

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