#72234326

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	#513082	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ #11408 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,d^2=egj,e^2=gi,f^2=h,g^2=ij,i^2=j,c^a=ch,c^b=ch,d^a=de,d^b=dij,d^c=dj,e^a=eg,e^b=ej,f^d=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234334, #72234355, #72260437, …
maximal normal subgroup of	#72234334, #72234355, #72260437, …
maximal quotient of	#72323126

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