#72284769

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=f,d^2=hj,e^2=i,b^a=bi,c^b=ch,d^a=dhi,d^b=di,d^c=dg,e^a=ef,e^b=egij,e^d=efj,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285517
maximal normal subgroup of	#72285517

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