#72323151

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂×#2263	C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122219 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,4,9,5,10,11,15,6,8,14,3,7,12,16,13)(17,18,22,30,21,32,43,61,25,34,49,33,26,29,24,53)(19,35,38,60,40,55,41,79,42,71,54,80,37,75,76,67)(20,28,45,63,44,31,23,50,46,64,48,70,47,62,52,66)(27,36,57,39,56,51,69,77,59,74,58,78,68,72,65,73);(17,19)(18,27)(20,39)(21,37)(22,41)(23,51)(24,54)(25,42)(26,40)(28,60)(29,56)(30,65)(31,67)(32,68)(33,69)(34,59)(35,63)(36,48)(38,43)(44,73)(45,74)(46,78)(47,77)(49,76)(50,75)(52,72)(53,57)(55,66)(58,61)(62,79)(64,80)(70,71);(1,3,11,2,7,15,4,12,6,9,16,8,5,13,14,10)(17,20,43,23,26,47,22,45,25,46,24,52,21,44,49,48)(18,28,61,50,29,62,30,63,34,64,53,66,32,31,33,70)(19,36,41,77,37,72,38,39,42,74,76,73,40,51,54,78)(27,55,69,80,68,35,57,79,59,75,65,60,56,71,58,67);(2,8)(3,9)(10,12)(13,15)(17,21,25,26)(18,29,34,32)(19,37,42,40)(20,44,46,47)(22,43,49,24)(23,48,52,45)(27,56,59,68)(28,62,64,31)(30,53,33,61)(35,55,71,75)(36,72,74,51)(38,76,54,41)(39,73,78,77)(50,63,66,70)(57,69,58,65)(60,79,80,67)

Polycyclic presentation:

2^11|a^2=e,c^2=eg,d^2=k,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^b=chjk,d^a=di,d^c=dik,e^b=ejk,f^a=fh,f^b=fhjk,f^c=fj,f^e=fk,g^b=gk,h^a=hj,h^b=hjk,h^c=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323152, #72323169
maximal normal subgroup of	#72323152, #72323169

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