#72304066

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 7163 conjugacy classes of subgroups; only the 2973 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=gi,d^2=j,e^2=j,b^a=bi,c^a=ch,c^b=cghi,d^a=di,d^c=dh,e^a=eg,e^b=eghij,e^c=egi,e^d=eij,f^c=fghij,f^e=fj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304145
maximal normal subgroup of	#72304145

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