#72323129

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	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#513009	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122219 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

754 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323133, #72323169
maximal normal subgroup of	#72323133, #72323169

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.5*F.7);;
gap> SetPower(coll, 4, F.11);;
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, 2, F.3*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*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, 3, F.6*F.10);;
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, 8, 3, F.8*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);