#72313844

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	#6628968	C₂
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3790 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:

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

Polycyclic presentation:

2^10|c^2=hij,f^2=i,g^2=j,b^a=bh,c^a=ch,c^b=cf,d^a=dj,d^b=dg,d^c=de,e^a=ej,e^b=eh,e^c=ej,f^b=fi,f^c=fi,g^a=gj,g^b=gj,g^c=ghj,g^d=gj,g^e=gj,h^d=hj

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313853, #72313950, #72313956
maximal normal subgroup of	#72313853, #72313950, #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, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
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, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*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);