#72265152

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18194
commutator, Frattini, cosocle	C₂×#97	C₂⁴
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|b^2=j,c^2=egj,e^2=hj,f^2=ij,g^2=j,h^2=j,i^2=j,b^a=bf,c^a=cf,c^b=ce,d^b=dhi,d^c=dgij,e^a=eg,e^b=eh,e^c=ej,e^d=ej,f^a=fi,f^b=fi,f^c=fgij,f^e=fj,g^a=gj,g^c=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265163, #72307737, #72307751
maximal normal subgroup of	#72265163, #72307737, #72307751

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