#72319992

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#365098
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=eij,d^2=j,e^2=j,f^2=i,g^2=j,i^2=j,b^a=bf,c^a=cg,c^b=cfi,d^a=dh,d^c=dj,e^b=ej,f^a=fi,f^b=fi,f^c=fij,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320043, #72343925, #72343946
maximal normal subgroup of	#72320043, #72343925, #72343946

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