#72338731

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72234638
commutator, Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
prime radical	#72311261	C₂
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

5641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|b^2=g,c^2=hij,d^2=hik,b^a=bh,c^a=cf,c^b=ce,d^a=dhik,d^b=dghijk,d^c=dhjk,e^a=ei,e^b=eik,e^c=ej,e^d=eijk,f^b=fi,f^d=fk,g^a=gjk,g^c=gik,g^d=gj,g^e=gk,g^f=gk,h^b=hjk,h^c=hk,h^d=hk,h^e=hk,i^b=ik,i^d=ik,j^c=jk

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338741, #72341003
maximal normal subgroup of	#72338741, #72341003

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