#72323070

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	C₂²×D₃₂	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

820 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:

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

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=chijk,d^a=dk,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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323084, #72323147, #72323164
maximal normal subgroup of	#72323084, #72323147, #72323164

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