#72287741

order	1024
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	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|c^2=h,d^2=f,e^2=fhij,f^2=h,g^2=ij,i^2=j,c^a=ch,c^b=chj,d^a=dh,d^b=dj,d^c=dg,e^b=ehj,e^d=egh,f^c=fj,f^e=fj,g^c=gi,g^d=gij,g^e=gij,i^c=ij,i^d=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287790, #72319058, #72319381
maximal normal subgroup of	#72287790, #72319058, #72319381

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