#72320101

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#106
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,d^2=fgj,h^2=i,i^2=j,b^a=bhij,d^a=dg,d^b=df,d^c=dj,e^a=ej,h^b=hi,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320153, #72320242, #72320500
maximal normal subgroup of	#72320153, #72320242, #72320500
maximal quotient of	#72323076

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