#72285676

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

2344 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fi,d^2=ghij,e^2=gj,c^a=cfi,c^b=cfj,d^a=df,d^b=dfh,d^c=dg,e^a=efhij,e^b=egh,e^c=efhi,e^d=ehij,f^d=fi,f^e=fij,g^a=gi,g^b=gj,g^c=gi,g^d=gj,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

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