#72341002

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

2301 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=fghj,d^2=fij,e^2=j,b^a=bgi,c^a=ch,c^b=cf,d^a=dfghj,d^b=dhj,d^c=dgh,e^a=egi,e^b=ej,e^c=eh,e^d=egi,f^a=fi,f^b=fij,f^c=fi,f^e=fj,g^c=gij,g^d=gj,h^a=hij,h^b=hj,h^c=hj,h^d=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341033, #72341091, #72341103
maximal normal subgroup of	#72341033, #72341091, #72341103

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