#72265280

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#23445
commutator, Frattini, cosocle	C₂²×#117	C₂⁴
socle	C₂²	#414622
prime radical	#609717	C₂²
LCS	G ▷ C₂²×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×#117 ▷ C₂³ ▷ C₂ ▷ C₁

4617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^11|a^2=ij,b^2=h,c^2=fij,d^2=hij,e^2=j,g^2=k,h^2=k,b^a=bg,c^a=cf,c^b=ce,d^a=dk,d^b=dhij,d^c=djk,e^a=ei,e^b=ej,f^b=fjk,g^b=gk,g^c=gij,g^e=gk,i^b=ik

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265475, #72308088
maximal normal subgroup of	#72265475, #72308088

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