#72313504

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	C₂×#3018	C₄
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3220 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)(11,16)(12,17)(13,26)(14,20)(15,19)(18,29)(21,30)(22,38)(23,32)(24,33)(25,36)(27,40)(28,34)(31,39)(35,37);(9,11)(10,16)(12,24)(13,27,29,37)(14,21)(15,23)(17,33)(18,35,26,40)(19,32)(20,30)(22,25,39,28)(31,34,38,36);(1,2,3,6,4,5,7,8)(9,12)(10,17)(11,21)(13,28,29,25)(14,15)(16,30)(18,34,26,36)(19,20)(22,27,39,37)(23,24)(31,40,38,35)(32,33);(9,13)(10,18)(11,22)(12,25)(14,28)(15,29)(16,31)(17,34)(19,26)(20,36)(21,37)(23,39)(24,27)(30,40)(32,38)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313518, #72313533, #72313911
maximal normal subgroup of	#72313518, #72313533, #72313911

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