#72320252

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₈
commutator	C₂²×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
socle	C₂³	C₂²×C₄×D₈
prime radical	#72320231	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×#41 ◁ C₂×C₈×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323148, #72323190
maximal normal subgroup of	#72323148, #72323190
maximal quotient of	#72321025

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