#72234397

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₃₂ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈×C₁₆	C₄²
commutator	C₄	C₂×C₈×C₃₂
Frattini, cosocle	C₈×C₃₂	C₂³
socle	C₂²	C₂×#7329
prime radical	C₂³	#7329
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=e,c^2=ik,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bik,c^a=ck,d^b=dk,e^a=ek

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234400
maximal normal subgroup of	#72234400

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