#72231988

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	C₂²×#1368
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂×#41	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #479943 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 96 points:

(1,2,4,9,6,10,15,22,5,8,14,20,16,21,26,30)(3,7,12,17,13,19,25,28,11,18,23,27,24,29,31,32)(33,34,36,44,38,46,57,78,40,48,58,47,41,66,39,65)(35,49,52,80,54,81,88,82,50,75,84,96,64,83,55,91)(37,59,60,79,61,93,62,73,51,77,56,92,63,95,86,45)(42,67,68,85,69,87,76,94,43,72,74,53,71,89,70,90);(1,3)(2,7)(4,12)(5,11)(6,13)(8,18)(9,17)(10,19)(14,23)(15,25)(16,24)(20,27)(21,29)(22,28)(26,31)(30,32)(35,50)(42,43)(49,75)(52,84)(53,85)(54,64)(55,88)(67,72)(68,74)(69,71)(70,76)(80,96)(81,83)(82,91)(87,89)(90,94);(2,8)(3,11)(9,20)(10,21)(12,23)(13,24)(22,30)(25,31)(33,35,37,53,39,52,62,87,41,54,63,90,58,88,60,72,40,50,51,85,57,84,86,89,38,64,61,94,36,55,56,67)(34,42,45,49,47,70,59,82,46,71,79,83,65,74,93,80,48,43,73,75,44,76,77,91,66,69,92,81,78,68,95,96);(33,35,51,85,39,52,86,89,41,54,61,94,58,88,56,67,40,50,37,53,57,84,62,87,38,64,63,90,36,55,60,72)(34,43,73,49,47,76,77,82,46,69,92,83,65,68,95,80,48,42,45,75,44,70,59,91,66,71,79,81,78,74,93,96)

Polycyclic presentation:

2^11|a^2=e,c^2=fik,d^2=fi,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^a=cf,c^b=ci,d^a=dfi,d^b=dk,d^c=dk,e^c=ejk,e^d=ejk,f^a=fh,f^e=fk,g^c=gk,g^d=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231993
maximal normal subgroup of	#72231993

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