#72286761

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	C₂×#13661	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4);(5,6,9,8)(7,10,12,11)(14,19)(15,25)(16,27)(17,29)(20,26)(21,31)(22,33)(23,41)(24,38)(28,36)(30,43)(32,34)(35,39)(37,42)(40,44);(2,4)(13,14,20,35,16,23,28,31,29,21,37,41,43,39,15,19,18,22,25,34,30,40,42,38,17,24,36,44,27,32,26,33);(5,7,8,11,9,12,6,10)(14,21,22,24)(15,26)(19,31,33,38)(20,25)(23,39,40,32)(28,42)(34,41,35,44)(36,37)

Polycyclic presentation:

2^11|b^2=i,c^2=ehk,d^2=g,e^2=hj,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=ce,d^c=dj,e^b=eh,e^d=ek,g^c=gk,h^b=hj,j^b=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287331, #72323148
maximal normal subgroup of	#72287331, #72323148
maximal quotient of	#72286964, #72321722

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