#72239907

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1438
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
socle	C₂²	#7555
prime radical	C₂⁵	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3139 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=d,c^2=e,d^2=i,f^2=ij,h^2=j,b^a=bg,c^a=cf,c^b=ch,e^a=ei,f^a=fij,f^b=fi,f^c=fj,g^c=gj,h^a=hij,h^b=hj,h^c=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239917
maximal normal subgroup of	#72239917
maximal quotient of	#72239909

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