#72271508

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁵
prime radical	C₂×2₋¹⁺⁴	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×Q₈² ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

4300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(9,10,15,20)(11,21,23,46)(12,26,28,49)(13,31,35,18)(14,36,39,19)(16,32,41,24)(17,37,44,30)(22,53,27,64)(25,59,56,52)(29,65,33,63)(34,47,38,50)(40,69,43,71)(42,58,48,61)(45,67,68,66)(51,60,72,55)(54,70,57,62);(9,11,15,23)(10,16,20,41)(12,27,28,22)(13,32,35,24)(14,25,39,56)(17,43,44,40)(18,46,31,21)(19,48,36,42)(26,62,49,70)(29,54,33,57)(30,60,37,55)(34,61,38,58)(45,51,68,72)(47,52,50,59)(53,65,64,63)(66,69,67,71);(9,12,15,28)(10,17,20,44)(11,22,23,27)(13,33,35,29)(14,37,39,30)(16,40,41,43)(18,45,31,68)(19,49,36,26)(21,51,46,72)(24,57,32,54)(25,60,56,55)(34,66,38,67)(42,70,48,62)(47,63,50,65)(52,64,59,53)(58,69,61,71);(1,2,4,6)(3,7,8,5)(11,23)(12,28)(16,41)(17,44)(21,46)(24,32)(25,56)(26,49)(29,33)(30,37)(42,48)(45,68)(52,59)(58,61)(63,65)(66,67);(1,3,4,8)(2,5,6,7)(12,28)(17,44)(22,27)(26,49)(29,33)(30,37)(40,43)(45,68)(51,72)(53,64)(54,57)(55,60)(62,70)(63,65)(66,67)(69,71)

Polycyclic presentation:

2^10|a^2=j,b^2=j,c^2=j,d^2=h,e^2=h,f^2=j,g^2=j,b^a=bf,c^a=cg,c^b=cj,d^b=dj,d^c=dj,e^c=ej,e^d=eh,f^a=fj,f^b=fj,f^c=fi,g^a=gj,g^b=gij,g^c=gj,g^f=gj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72271515
maximal subgroup of	#72271515
maximal normal subgroup of	#72271515
maximal quotient of	#72271541

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