#72257579

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7557
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂⁵	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

2161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=f,d^2=i,e^2=ij,h^2=j,b^a=bg,c^a=cd,c^b=ce,d^b=di,e^a=ej,f^a=fi,f^b=fij,g^c=gij,h^c=hi

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257599, #72316688, #72317025
maximal normal subgroup of	#72257599, #72316688, #72317025
maximal quotient of	#72257764

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