#72268618

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#3549
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	#60045
prime radical	C₂⁵	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,d^2=hij,e^2=hj,g^2=j,b^a=bf,c^a=ch,c^b=chi,d^a=dg,d^b=dhij,e^a=ehj,e^b=egj,e^d=ej,g^a=gj,g^b=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72268621
maximal subgroup of	#72268619, #72268621
maximal normal subgroup of	#72268619, #72268621
maximal quotient of	#72268636

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