#72285304

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	#36714	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2348 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)(3,14)(4,20)(5,27)(9,32)(12,13)(16,19)(17,46)(21,42)(22,50)(23,44)(24,30)(25,26)(34,36)(35,58)(39,57)(48,61)(49,60)(55,59)(56,62);(1,2,5,28)(3,15,18,32)(4,21,55,33)(6,8,27,11)(7,13,23,43)(9,14,45,47)(10,26,42,30)(12,44,41,31)(16,40,29,34)(17,49,64,62)(19,54,53,36)(20,39,24,38)(22,58,63,56)(25,57,59,37)(35,52,48,46)(50,60,51,61);(1,3)(2,9,8,32)(4,22,20,46)(5,29)(6,14)(7,16)(10,35,38,60)(11,40,28,54)(12,34,13,36)(15,43,45,41)(17,26,50,25)(18,23)(19,31)(21,56,57,62)(24,51,55,64)(27,53)(30,52,59,63)(33,49,37,58)(39,48,42,61)(44,47);(3,16)(5,23)(9,34)(10,37)(11,41)(14,19)(15,45)(17,46)(21,42)(22,50)(24,30)(27,44)(28,43)(32,36)(33,38)(35,58)(39,57)(40,54)(48,62)(49,60)(51,64)(52,63)(55,59)(56,61);(1,4,23,24)(2,10,28,21)(3,17,47,64)(5,30,7,25)(6,26,44,59)(8,33,11,42)(9,35,40,61)(12,37,41,39)(13,38,43,57)(14,46,18,52)(15,48,36,60)(16,22,53,63)(19,50,29,51)(20,27,55,31)(32,49,54,56)(34,58,45,62)

Polycyclic presentation:

2^10|b^2=fg,c^2=f,e^2=ghj,b^a=bf,c^a=cfi,c^b=cgh,d^b=dfhj,d^c=dfj,e^a=efhj,e^b=egij,e^c=efh,e^d=ehj,f^b=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285772
maximal normal subgroup of	#72285772

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