#72322349

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#2177
commutator	C₂×C₁₆	C₂⁴×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁵
prime radical	#72232017	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17335 ◁ #479917 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 80 points:

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

Polycyclic presentation:

2^11|c^2=f,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bh,c^a=cj,c^b=ch,d^b=dk,e^a=eg,e^c=ejk,f^a=fk,f^e=fk,g^a=gi,g^c=gk,i^a=ij,j^a=jk

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322589, #72323115, #72323123, …
maximal normal subgroup of	#72322589, #72323115, #72323123, …

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