#72233155

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#393934
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂×D₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 9350 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^11|b^2=gjk,c^2=g,b^a=bg,c^a=cg,c^b=cjk,d^a=de,d^b=dh,d^c=def,e^b=ei,e^c=eij,f^a=fj,f^b=fi,f^c=fj,f^d=fk,f^e=fk,g^d=gik,h^a=hk,h^b=hi,h^c=hk,j^d=jk

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233213
maximal normal subgroup of	#72233213

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