#72284584

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	#245279	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2247 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)(10,19)(12,20)(13,21)(14,25)(15,22)(16,23)(17,24)(18,26)(27,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,48)(38,42)(41,49)(43,50)(44,51)(45,52)(46,60)(47,54)(53,57)(55,61)(56,58)(59,62)(63,64);(1,3,6,15)(2,7,11,22)(4,12,17,28)(5,13)(8,20,24,31)(9,25)(10,21)(14,19)(16,29)(18,27)(23,30)(26,32)(33,35)(34,39)(36,44,47,57)(37,45)(38,43)(40,53,54,51)(41,56)(42,50)(46,59,61,64)(48,52)(49,58)(55,62,60,63);(1,4)(2,8)(3,12)(5,18)(6,17)(7,20)(9,23)(10,26)(11,24)(13,27)(14,29)(15,28)(16,19)(21,32)(22,31)(25,30)(35,43)(39,50)(40,54)(44,57)(45,58)(46,61)(52,56)(62,63);(1,5,6,19)(2,9)(3,13)(4,16)(7,21,22,25)(8,23,24,26)(10,11)(12,27,28,29)(14,15)(17,18)(20,30)(31,32)(33,36,38,47)(34,40)(35,44)(37,46,49,61)(39,51,50,53)(41,55)(42,54)(43,57)(45,59)(48,60)(52,62,56,63)(58,64);(1,5)(2,10,11,9)(3,14,15,13)(4,16,17,18)(6,19)(7,21)(8,23)(12,27)(20,30,31,32)(22,25)(24,26)(28,29)(33,37,38,49)(34,41)(35,45)(36,46,47,61)(39,52,50,56)(40,55)(42,48)(43,58)(44,59)(51,62,53,63)(54,60)(57,64)

Polycyclic presentation:

2^10|b^2=fij,d^2=gi,e^2=gj,b^a=bf,c^a=cf,c^b=cj,d^a=dfi,d^b=dfg,d^c=dfgh,e^a=efh,e^b=efghi,e^c=ef,e^d=eij,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gij,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	#72284848
maximal normal subgroup of	#72284848

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