#72285426

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

2444 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fhj,e^2=i,c^a=cg,c^b=ch,d^a=dhij,d^b=dgj,d^c=dg,e^a=eij,e^b=ehij,e^c=ef,e^d=eij,f^a=fi,f^b=fj,f^d=fi,g^c=gi,g^d=gj,g^e=gi,h^c=hj,h^d=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185
maximal normal subgroup of	#72286185

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