#72307172

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	C₂×C₄²	C₂⁵
socle	C₂²	C₂×#2535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3821 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,e^2=hj,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,c^b=cgh,d^a=di,e^a=ehj,e^b=ej,e^c=ei,f^a=fi,f^b=fj,f^c=fi,h^a=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307175, #72307250, #72308812
maximal normal subgroup of	#72307175, #72307250, #72308812

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