#72286256

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	#245285	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=j,c^2=gj,d^2=g,e^2=gh,b^a=bf,c^a=cg,c^b=ch,d^a=dg,d^b=dfj,d^c=di,e^a=efj,e^b=efhi,e^c=efg,e^d=ehj,f^c=fj,f^d=fi,f^e=fij,g^b=gi,g^e=gj,h^a=hij,h^c=hi,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, 1, F.10);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*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, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
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);