#72342662

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×#22670
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	C₂²×#509	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #183449 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5279 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)(3,8);(2,3)(4,7)(6,8)(10,16,18,20)(11,12)(13,14)(17,23,19,24)(21,22);(1,2,4,3,5,6,7,8)(9,10)(11,20)(12,23)(13,19)(14,17)(15,18)(16,22)(21,24);(2,6)(3,8)(9,11)(10,17)(12,13)(14,21)(15,22)(18,19)(23,24);(1,3,5,8)(2,7,6,4)(10,16,18,20)(11,12)(13,14)(17,23,19,24)(21,22)

Polycyclic presentation:

2^11|b^2=f,c^2=ghj,e^2=fij,g^2=i,h^2=jk,i^2=k,j^2=k,c^a=cij,c^b=cg,d^b=dj,d^c=dhi,e^a=eij,e^b=eghik,e^c=egik,e^d=ei,f^c=fk,g^b=gi,g^c=gj,g^d=gj,g^e=gi,h^b=hjk,h^c=hj,h^d=hj,h^e=hjk,i^b=ik,i^c=ik,i^d=ik,i^e=ik,j^b=jk,j^c=jk,j^d=jk,j^e=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, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.6*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);