#72323096

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 ◁ C₂×#7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

733 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=k,c^2=eg,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=ck,c^b=chj,d^a=di,d^b=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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323108, #72323169
maximal normal subgroup of	#72323108, #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);;
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.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, 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);