#72307246

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	C₂×#2535
prime radical	#6628994	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5831 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)(9,10,13,19,15,17,14,30)(11,20,23,31,22,35,24,16)(12,25,27,33,29,40,28,18)(21,34,37,39,26,32,38,36);(1,3)(2,5)(4,7)(6,8)(16,31)(20,35)(32,34)(36,39);(3,7)(5,8)(9,11,15,22)(10,16)(12,26,29,21)(13,23,14,24)(17,31)(18,34)(19,20)(25,39)(27,38,28,37)(30,35)(32,33)(36,40);(2,6)(5,8)(10,17)(16,31)(18,33)(19,30)(20,35)(25,40)(32,34)(36,39);(9,12,14,28,15,29,13,27)(10,18,19,25,17,33,30,40)(11,21,24,38,22,26,23,37)(16,32,20,36,31,34,35,39)

Polycyclic presentation:

2^10|a^2=h,c^2=hij,e^2=i,h^2=j,i^2=j,b^a=bf,c^a=ci,c^b=cg,d^a=dghij,e^a=ei,e^c=ej,f^c=fj,g^a=gj,i^a=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307248, #72307402, #72307438
maximal normal subgroup of	#72307248, #72307402, #72307438

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