#72257501

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1437
commutator	C₄²	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂×#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

3655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=h,e^2=i,f^2=j,h^2=j,b^a=bf,c^a=ce,d^b=dj,d^c=di,e^a=ei,e^b=ej,f^a=fj,f^c=fj,g^a=gj,h^a=hi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257538
maximal normal subgroup of	#72257538
maximal quotient of	#72257756

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