#72280943

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	C₂×C₄⋊C₄	C₂⁵
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

5882 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=j,c^2=h,d^2=ij,f^2=i,g^2=j,b^a=bh,c^a=ch,d^a=df,d^b=dg,d^c=dfij,e^b=ehj,e^c=ej,f^a=fi,f^b=fh,f^c=fij,f^d=fi,g^a=gh,g^b=gj,g^c=ghj,g^d=gj,g^e=gj,g^f=gj,h^d=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281530, #72282096
maximal normal subgroup of	#72281530, #72282096

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