#72287711

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂×#61427	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,9,6,8,14,11)(3,7,12,13,10,15,16,4)(17,18,21,27,22,25,36,26,23,29,37,28,20,33,35,44)(19,24,32,40,30,42,46,45,31,43,47,41,34,38,48,39);(3,10)(4,13)(7,15)(12,16)(17,19)(18,24)(20,34)(21,32)(22,30)(23,31)(25,42)(26,45)(27,40)(28,41)(29,43)(33,38)(35,48)(36,46)(37,47)(39,44);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,13)(14,16)(17,20,23,22)(18,25,29,33)(19,30,31,34)(21,35,37,36)(24,38,43,42)(26,28,44,27)(32,46,47,48)(39,41,45,40);(2,8)(3,10)(9,11)(12,16)(17,19)(18,24)(20,34)(21,32)(22,30)(23,31)(25,42)(26,45)(27,40)(28,41)(29,43)(33,38)(35,48)(36,46)(37,47)(39,44);(1,4,14,15,6,13,5,7)(2,3,11,16,8,10,9,12)(18,26,29,44)(19,31)(20,22)(21,36)(24,39,43,45)(25,27,33,28)(32,48)(35,37)(38,40,42,41)(46,47)

Polycyclic presentation:

2^10|a^2=f,c^2=j,e^2=fghj,f^2=h,g^2=i,h^2=j,i^2=j,c^a=cj,c^b=chi,d^a=dhij,d^c=dhi,e^a=eg,e^b=ehi,e^c=ej,e^d=ej,f^e=fi,g^a=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287726, #72287751, #72320918
maximal normal subgroup of	#72287726, #72287751, #72320918

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