#72340979

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

43203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 40 points:

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

Polycyclic presentation:

2^10|d^2=h,e^2=hi,f^2=j,g^2=ij,d^b=di,d^c=dij,e^b=eij,e^c=ej,e^d=ehij,f^a=fj,f^b=fj,f^d=fhj,f^e=fj,g^a=gij,g^b=gij,g^d=gij,g^e=ghj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340982
maximal subgroup of	#72340982, #72341169
maximal normal subgroup of	#72340982, #72341169

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