#72249773

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#24338
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #60191 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4669 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=h,d^2=hj,e^2=gij,g^2=i,i^2=j,b^a=bh,c^a=ch,d^a=dj,d^b=dhj,d^c=dhj,e^a=ef,e^b=efgij,e^c=efij,e^d=ef,f^a=fj,f^d=fj,g^b=gi,g^c=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249843
maximal normal subgroup of	#72249843
maximal quotient of	#72256845

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