#72234331

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#357
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	#7623990	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ C₁₆×#41 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234334, #72322276, #72322477
maximal normal subgroup of	#72234334, #72322276, #72322477
maximal quotient of	#72323093

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