#72285371

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	#245273	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2261 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)(3,8)(4,16)(5,10)(6,11)(7,12)(9,19)(13,25)(14,23)(15,24)(17,22)(18,27)(20,28)(21,26)(29,32)(30,31)(37,50)(53,57)(55,56)(58,60);(1,3,7,15)(2,8,12,24)(4,17,19,13)(5,14)(6,20)(9,25,16,22)(10,28)(11,23)(18,29)(21,30)(26,32)(27,31)(33,34)(35,43)(36,47)(37,51,50,41)(38,42)(39,46)(40,49)(44,54)(45,53,61,57)(48,55,62,56)(52,59)(58,63,60,64);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,27)(11,26)(12,16)(14,30)(15,17)(20,29)(23,31)(24,25)(28,32)(33,35,38,46)(34,39,42,43)(36,44,49,59)(37,45,50,61)(40,54,47,52)(41,53,51,57)(48,58,62,60)(55,64,56,63);(1,5,7,6)(2,10)(3,14)(4,18,19,21)(8,23,24,28)(9,26)(11,12)(13,29)(15,20)(16,27)(17,30)(22,31,25,32)(33,36)(34,40,42,47)(35,44)(37,48,50,62)(38,49)(39,52,43,54)(41,56)(45,60,61,58)(46,59)(51,55)(53,63)(57,64);(1,6,7,5)(2,11)(3,14)(4,18)(8,23,24,28)(9,27,16,26)(10,12)(13,29,17,30)(15,20)(19,21)(22,32)(25,31)(33,37,38,50)(34,41,42,51)(35,45)(36,48)(39,53)(40,55)(43,57)(44,58,59,60)(46,61)(47,56)(49,62)(52,63,54,64)

Polycyclic presentation:

2^10|b^2=fi,c^2=j,d^2=ghi,e^2=gij,b^a=bf,c^a=cfi,c^b=cij,d^a=dfi,d^b=dfg,d^c=dgh,e^a=efhij,e^b=efghi,e^c=efj,e^d=eh,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gij,g^d=gj,h^b=hj,h^c=hij,h^d=hj,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.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.7*F.9*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.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
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.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, 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, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);