#72341214

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
prime radical	#251883	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2284 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=fg,d^2=j,e^2=ghj,b^a=bf,c^a=cfh,c^b=cfh,d^a=dj,d^b=dh,d^c=dfh,e^a=efhj,e^b=efi,e^c=efghij,e^d=ehi,f^b=fi,f^c=fij,f^e=fij,g^a=gi,g^b=gi,g^d=gj,g^e=gj,h^b=hj,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341219
maximal normal subgroup of	#72341219

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