#72231730

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72231727
commutator, Frattini, cosocle	C₂×#131	C₂⁵
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

At least 17547 conjugacy classes of subgroups; only the 1026 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(3,12)(9,16)(13,28)(15,24)(17,27)(18,31)(21,25)(26,32);(1,2)(3,12)(4,8)(5,10)(6,19)(7,11)(9,16)(13,26)(14,20)(15,25)(17,27)(18,31)(21,24)(22,30)(23,29)(28,32);(3,13,18,32)(4,19)(5,20)(6,8)(9,24,27,25)(10,14)(12,26,31,28)(15,16,21,17)(22,23)(29,30);(1,2)(3,13,18,32)(4,6)(5,14)(7,11)(8,19)(9,25,27,24)(10,20)(12,28,31,26)(15,16,21,17)(22,29)(23,30);(1,3,14,25,8,17,30,26,7,18,29,24,6,16,10,28)(2,9,23,13,4,12,5,21,11,27,20,32,19,31,22,15)

Polycyclic presentation:

2^11|c^2=ik,d^2=ik,e^2=fgjk,g^2=j,h^2=k,j^2=k,c^a=cik,d^b=dik,e^a=ef,e^b=eh,e^c=eg,e^d=efgjk,f^b=fi,f^c=fk,g^b=gik,g^c=gj,g^d=gj,h^a=hik,h^b=hk,h^c=hi,h^f=hk,h^g=hk,i^e=ik,j^c=jk,j^d=jk

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231730
maximal normal subgroup of	C₂×#72231730
maximal quotient of	#72257273, #72257323, #72314138, …
automorphism group of	#1384, #1385, #2161, …

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