#72341178

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂²×#560	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2942 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=i,d^2=f,e^2=j,b^a=bj,c^a=cij,c^b=ch,d^a=dij,d^b=df,d^c=dfg,e^a=ei,e^b=efghij,e^d=eh,f^c=fi,f^e=fj,g^b=gij,g^c=gi,g^d=gi,g^e=gj,h^b=hi,h^d=hj

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341183, #72341186
maximal subgroup of	#72341180, #72341183
maximal normal subgroup of	#72341180, #72341183

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