#72341401

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5850 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,33)(20,30)(21,34)(22,29)(23,25)(24,31)(26,27)(28,32)(35,36)(37,39)(38,40);(3,4)(6,7)(9,10)(11,19)(12,23)(13,26)(14,15)(16,21)(17,25)(18,27)(20,35)(22,34)(24,31)(28,32)(29,33)(30,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(10,15)(13,27)(16,29)(17,23)(19,34)(24,28)(30,39)(35,40);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,20)(13,21,27,33)(14,22)(15,29)(17,30)(18,19,26,34)(23,39)(24,36,28,38)(25,37)(31,40,32,35);(2,8)(3,4)(9,12,14,25)(10,17,15,23)(11,20,22,37)(13,28,27,24)(16,30,29,39)(18,32,26,31)(19,35,34,40)(21,38,33,36)

Polycyclic presentation:

2^10|c^2=hj,d^2=fhj,e^2=j,g^2=j,b^a=bi,c^a=cj,c^b=ch,d^a=df,d^b=dg,d^c=df,e^a=ehj,e^b=eij,e^c=ehj,e^d=ehj,f^b=fi,g^a=gi,g^b=gj,g^c=gij,g^d=gij,g^e=gj,g^f=gj,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341430
maximal normal subgroup of	#72341430

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