#72342659

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#23001
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72342611	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #183449 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,6)(3,7);(2,7)(3,6)(4,8)(10,15)(11,17)(12,19)(14,24)(16,20)(18,23)(21,22);(9,10,13,17)(11,14,15,24)(12,20)(18,19)(22,23);(1,2)(3,4)(5,6)(7,8)(9,11,14,17,13,15,24,10)(12,21,20,18,23,16,22,19);(1,3,4,2,5,7,8,6)(9,12,15,19,24,20,17,21,13,23,11,18,14,22,10,16)

Polycyclic presentation:

2^11|c^2=f,d^2=fjk,e^2=fghijk,g^2=i,h^2=jk,j^2=k,c^b=cjk,d^a=dik,d^b=dg,d^c=dj,e^a=eik,e^b=eghj,e^c=ehijk,e^d=egik,f^e=fk,g^b=gi,g^c=gk,g^d=gik,h^b=hj,h^c=hjk,h^d=hk,h^e=hk,j^b=jk,j^c=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×AGL(1,ℤ/16)
maximal normal subgroup of	AGL(1,ℤ/8)×AGL(1,ℤ/16)

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