#72225107

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225106
commutator	C₂×C₁₆	C₂²×C₄²
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×#3415	C₂×C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479908 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,4,13,6,10,22,41,7,12,23,34,29,40,5,14,8,15,25,9,30,35,28,36,31,38,58,37,27,45,26,32)(3,16,17,44,19,46,49,63,20,47,50,59,55,64,53,43,21,11,24,33,56,39,52,60,57,42,51,61,54,62,18,48);(2,9,12,34,15,13,38,37)(4,22,25,28)(5,23,26,58)(6,29)(7,31)(10,32,40,36,35,14,45,41)(11,33,42,59,16,44,47,61)(17,49,24,52)(18,51,53,50)(19,55)(20,57)(27,30)(39,48,62,60,46,43,64,63)(54,56);(2,10,38,45,15,35,12,40)(4,23,25,58)(5,22,26,28)(6,30)(9,32,37,41,13,14,34,36)(11,39,47,64,16,46,42,62)(17,50,24,51)(18,52,53,49)(19,56)(27,29)(33,48,61,63,44,43,59,60)(54,55);(1,3)(2,11)(4,24)(5,18)(6,19)(7,20)(8,21)(9,33)(10,39)(12,42)(13,44)(14,43)(15,16)(17,25)(22,52)(23,51)(26,53)(27,54)(28,49)(29,55)(30,56)(31,57)(32,48)(34,59)(35,46)(36,60)(37,61)(38,47)(40,62)(41,63)(45,64)(50,58)

Polycyclic presentation:

2^11|a^2=f,b^2=e,c^2=eh,e^2=h,f^2=i,g^2=ijk,i^2=j,j^2=k,b^a=bg,c^a=cijk,d^a=dh,e^a=ejk,f^b=fi,f^c=fj,f^d=fk,f^e=fk,g^b=gijk,g^c=gjk,g^e=gk,h^a=hk,i^b=ij,i^c=ik,j^b=jk

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225134, #72225136, C₂×#72225107, …
maximal normal subgroup of	#72225134, #72225136, C₂×#72225107, …
maximal quotient of	C₂×#72225107

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