#72282587

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#116855
commutator	C₂×#131	C₂²×C₄
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂×#585	D₄
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dg,c^2=ehij,e^2=i,f^2=ij,h^2=j,i^2=j,b^a=bg,c^a=ce,c^b=cf,d^c=dj,e^a=ei,e^b=ei,f^a=fh,f^b=fh,g^c=ghi,g^f=gj,h^a=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72341787, #72341830

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