#72227051

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#128900
commutator	C₄×C₁₆	C₂×C₄²
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂×#12515	C₂²
LCS	G ▷ C₄×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=f,d^2=ghi,e^2=h,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cij,c^b=ck,d^a=dg,d^b=dghj,d^c=dk,e^b=ehj,f^a=fk,g^a=gi,g^b=gi,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227164
maximal normal subgroup of	#72227164
maximal quotient of	#72227058

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