#72322875

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 ◁ #122214 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

822 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=egi,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=ci,c^b=chjk,d^c=dk,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

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322903, #72323144, #72323152
maximal normal subgroup of	#72322903, #72323144, #72323152

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*F.9);;
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, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*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);