#72307361

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	C₂×#2398
prime radical	#60444	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3799 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)(3,5)(4,6)(7,8)(9,10,13,19,14,18,28,15)(11,20,22,16,24,31,23,34)(12,25,29,17,26,32,30,36)(21,35,38,40,27,33,39,37);(1,3)(2,5)(4,7)(6,8)(10,15,18,19)(12,26)(16,20,34,31)(17,32,36,25)(21,27)(29,30)(33,37,35,40)(38,39);(2,6)(3,7)(9,11)(10,16,18,34)(12,27)(13,23)(14,24)(15,31,19,20)(17,35,36,33)(21,26)(22,28)(25,40,32,37)(29,38)(30,39);(2,6)(3,7)(9,12,28,29,14,26,13,30)(10,17,19,25,18,36,15,32)(11,21,22,39,24,27,23,38)(16,33,20,37,34,35,31,40);(2,6)(5,8)(10,18)(15,19)(16,34)(17,36)(20,31)(25,32)(33,35)(37,40)

Polycyclic presentation:

2^10|a^2=f,b^2=hj,c^2=hj,d^2=ij,f^2=j,g^2=j,i^2=j,b^a=bfh,c^a=cg,c^b=cgij,d^a=dg,d^b=dgi,d^c=dj,e^a=eghij,f^c=fj,f^d=fj,h^a=hj,i^a=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307402
maximal normal subgroup of	#72307402

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