#72323121

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#2177
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	C₂×#33733	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₈×M₁₆ ◁ C₂×#11408 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,10,8,9,12,18,17)(7,11,16,14,15,13,19,20)(22,26)(23,31,33,48)(24,35)(25,39)(28,42,32,43)(29,44)(30,41)(34,45,47,51)(36,46)(37,49,52,40)(38,50);(5,7)(6,11)(8,14)(9,15)(10,16)(12,13)(17,20)(18,19);(1,2)(3,4)(5,6,10,8,9,12,18,17)(7,13,16,20,15,11,19,14)(21,22,27,26)(24,36,38,29)(25,30,41,39)(31,48)(35,44,50,46)(37,52)(42,43)(45,51);(2,4)(5,8,10,12,9,17,18,6)(7,14,16,13,15,20,19,11)(21,23,26,32,27,33,22,28)(24,37,36,51,38,52,29,45)(25,40,39,34,41,49,30,47)(31,44,42,50,48,46,43,35)

Polycyclic presentation:

2^11|a^2=g,c^2=gk,d^2=gj,e^2=k,f^2=hjk,g^2=i,h^2=jk,j^2=k,c^a=ck,c^b=ci,d^a=df,d^c=de,e^a=ek,e^d=ek,f^a=fh,f^c=fk,f^d=fh,g^d=gk,h^a=hj,h^d=hj,j^a=jk,j^d=jk

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323148
maximal normal subgroup of	#72323148
maximal quotient of	#72323123

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