#72227087

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12316
commutator, Frattini, cosocle	C₈×C₁₆	C₂⁴
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆×D₈ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,9)(8,11)(10,12)(13,14)(15,16);(3,11)(4,12)(6,8)(7,10)(13,16)(14,15)(18,23)(19,27)(20,29)(21,33)(24,36)(25,39)(26,40)(28,32)(30,46)(31,43)(34,45)(35,37)(38,47)(41,42)(44,48);(2,6)(3,11)(4,12)(7,15)(8,10)(9,14)(13,16)(17,18)(19,28)(20,32)(21,35)(22,24)(23,27)(25,29)(26,33)(30,47)(31,44)(34,48)(36,43)(37,46)(38,45)(39,41)(40,42);(18,24)(23,36)(25,38)(26,35)(28,44)(32,48)(37,40)(39,47)

Polycyclic presentation:

2^11|e^2=gi,f^2=hj,g^2=ik,h^2=j,i^2=k,c^a=cf,c^b=ce,d^c=dk,e^a=eh,e^b=eg,e^c=eg,f^a=fh,f^b=fh,f^c=fh,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^b=ik,i^c=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227164
maximal normal subgroup of	#72227164
maximal quotient of	#72227092

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