#72267307

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄²

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

17273 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|e^2=j,f^2=j,c^a=cg,c^b=cgh,d^a=dghi,d^b=dhij,e^c=ei,e^d=ej,f^c=fj,f^d=fij,f^e=fj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267325
Fitting subgroup of	#72267325
maximal subgroup of	#72267325
maximal normal subgroup of	#72267325
maximal quotient of	#72267321

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