#72308286

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1786
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#10577535	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=efhj,d^2=ij,e^2=h,h^2=j,i^2=j,b^a=bg,c^a=ce,c^b=cg,d^a=dh,d^c=dhij,e^a=eh,e^b=ehij,e^c=ej,f^c=fj,g^c=ghij,h^a=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308298, #72308521, #72308568
maximal normal subgroup of	#72308298, #72308521, #72308568

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