#72313907

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	C₂×C₄	C₂×B(4,𝔽₂)
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
socle	C₂²	C₂×#1889
prime radical	C₂²×H(5,𝔽₂)	C₂×C₄
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3406 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:

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

Polycyclic presentation:

2^10|a^2=hj,b^2=f,f^2=i,g^2=j,b^a=bhi,c^b=cj,d^a=dg,d^b=de,e^a=eh,e^b=ej,f^d=fj,g^b=gh,g^d=gj,g^e=gj,h^d=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313911, #72313956
maximal normal subgroup of	#72313911, #72313956

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