#72232014

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	#513080	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=egj,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=bh,c^a=ce,c^b=cj,d^c=di,e^a=eg,e^d=ej,f^c=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232026, #72234334, #72234355, …
maximal normal subgroup of	#72232026, #72234334, #72234355, …
maximal quotient of	#72321095, #72322835

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