#72322454

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1333
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#7623249	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #479899 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1712 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:

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

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,c^a=ci,c^b=ci,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

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322541, #72323133, #72323164
maximal normal subgroup of	#72322541, #72323133, #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, 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.9);;
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);