#72320143

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#392992
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#7622976	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #60058 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2581 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)(3,15)(4,20)(5,11)(6,30)(7,33)(8,12)(9,14)(10,25)(13,16)(17,31)(18,43)(19,32)(21,44)(22,53)(23,41)(24,47)(26,49)(27,40)(28,39)(29,37)(34,36)(35,56)(38,45)(42,48)(46,54)(50,59)(51,55)(52,62)(57,60)(58,61)(63,64);(1,3,16,12,7,19,32,33,8,13,15,2,9,30,6,14)(4,21,42,47,25,17,31,10,24,48,44,20,34,41,23,36)(5,18,45,37,28,50,59,39,29,38,43,11,27,49,26,40)(22,52,62,53,57,46,58,35,55,64,63,51,56,61,54,60);(1,4)(2,10,12,36)(3,17,13,41)(5,22)(6,31)(7,25)(8,24)(9,34)(11,35,37,60)(14,20,33,47)(15,42)(16,44)(18,46,38,61)(19,48,30,21)(23,32)(26,58)(27,56)(28,57)(29,55)(39,53,40,51)(43,62)(45,63)(49,52,50,64)(54,59);(1,5)(2,11)(3,18)(4,22)(6,26)(7,28)(8,29)(9,27)(10,35)(12,37)(13,38)(14,40)(15,43)(16,45)(17,46)(19,50)(20,51)(21,52)(23,54)(24,55)(25,57)(30,49)(31,58)(32,59)(33,39)(34,56)(36,60)(41,61)(42,62)(44,63)(47,53)(48,64)

Polycyclic presentation:

2^10|b^2=fi,c^2=e,f^2=ij,h^2=j,i^2=j,b^a=bf,c^a=ch,c^b=cij,d^a=dg,e^a=ej,e^b=ej,f^a=fi,f^c=fj,g^c=gj,h^a=hj,h^d=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320238, #72320817, #72343971
maximal normal subgroup of	#72320238, #72320817, #72343971

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