#72302992

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#106766
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

2996 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=g,c^2=hj,e^2=j,f^2=j,b^a=be,c^a=cdefj,d^b=dh,d^c=dh,e^a=ehj,e^b=ei,e^c=eh,e^d=ej,f^b=fij,g^a=gij,g^e=gj,g^f=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303005, #72303117
maximal normal subgroup of	#72303005, #72303117

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