#72322417

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1333
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂×#3415	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #479899 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2041 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:

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

Polycyclic presentation:

2^11|a^2=e,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,d^c=dk,e^b=ejk,f^a=fh,f^b=fhjk,f^e=fk,g^b=gk,h^a=hj,h^b=hjk,j^a=jk,j^b=jk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322541, #72323120, #72323133
maximal normal subgroup of	#72322541, #72323120, #72323133

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, 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, 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, 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, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);