#72265677

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#195202
commutator	C₂×#97	C₂³×C₄
Frattini, cosocle	C₂²×#97	C₂⁴
prime radical	#72265625	C₂
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5447 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)(7,13)(9,15)(10,16)(18,23,25,37)(20,30)(21,33)(24,36,39,47)(26,31,34,38)(28,43)(29,42)(32,35)(40,44,45,48)(41,46);(1,2)(3,9)(4,13)(5,8)(6,10)(7,12)(11,15)(14,16)(17,18)(19,24)(20,31)(21,34)(22,25)(23,35)(26,33)(27,39)(28,44)(29,45)(30,37)(32,38)(36,46)(40,42)(41,48)(43,47);(1,3)(2,6,8,14)(4,10)(5,11)(7,9,13,15)(12,16)(20,32)(24,39)(28,41)(30,35)(36,47)(40,45)(43,46)(44,48);(17,19)(18,24)(20,28)(21,29)(22,27)(23,36)(25,39)(26,40)(30,43)(31,44)(32,41)(33,42)(34,45)(35,46)(37,47)(38,48)

Polycyclic presentation:

2^11|a^2=hi,c^2=g,e^2=i,f^2=j,j^2=k,b^a=bf,c^b=ce,d^c=dhi,e^a=eh,e^b=ei,e^d=ek,f^a=fj,f^b=fjk,f^c=fhk,f^e=fk,g^b=gi,h^b=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265735, #72308022
maximal normal subgroup of	#72265735, #72308022

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