#72338809

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#251827
commutator, Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#72210559	C₂
LCS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

5783 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=e,d^2=k,f^2=jk,g^2=j,b^a=bg,c^a=ch,c^b=cj,d^a=df,d^b=de,d^c=de,e^a=eh,f^b=fi,f^c=fh,f^d=fjk,f^e=fjk,g^b=ghj,g^c=gj,g^d=ghik,g^e=gj,h^b=hj,h^d=hjk,i^b=ijk,i^c=ij,i^d=ijk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338830, #72338994, #72340564
maximal normal subgroup of	#72338830, #72338994, #72340564

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