#72322996

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	#2215	C₂×C₈
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

715 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=k,c^2=egk,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^b=chjk,d^b=di,d^c=di,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	#72323032, #72323169
maximal normal subgroup of	#72323032, #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, 2, F.11);;
gap> SetPower(coll, 3, F.5*F.7*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, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
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);