#72281069

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

5989 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)(6,8)(9,10)(11,15)(12,16)(13,26)(14,19)(17,24)(18,25)(20,29)(21,32)(22,35)(23,31)(27,30)(28,33)(34,36)(37,39)(38,40);(2,6)(4,7)(5,8)(9,11,14,23)(10,15,19,31)(12,20,25,22)(13,27)(16,29,18,35)(17,32)(21,36)(24,38)(26,39)(28,37)(30,33)(34,40);(2,6)(4,7)(5,8)(9,12)(10,16)(11,20)(13,24,28,36)(14,25)(15,29)(17,26,34,33)(18,19)(21,27,38,37)(22,23)(30,32,39,40)(31,35);(1,2,3,5)(4,8,7,6)(9,13)(10,17)(11,21)(12,24)(14,28)(15,30)(16,26)(18,33)(19,34)(20,27)(22,37)(23,38)(25,36)(29,32)(31,39)(35,40);(9,12,14,25)(10,18,19,16)(11,22,23,20)(13,24,28,36)(15,29,31,35)(17,33,34,26)(21,37,38,27)(30,32,39,40)

Polycyclic presentation:

2^10|b^2=hj,c^2=h,d^2=i,e^2=j,g^2=i,b^a=bh,c^a=ch,d^a=df,d^b=dfgi,d^c=dghj,e^a=ehj,e^b=ej,e^c=ehj,f^b=fh,f^c=fh,f^e=fj,g^a=gh,g^b=ghi,g^c=gij,g^d=gij,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, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.9);;
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.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
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, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);