#72249781

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×#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#10576634	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #60140 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4723 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)(3,9)(4,18)(5,24)(6,12)(7,11)(8,13)(10,22)(14,29)(15,36)(16,33)(17,32)(19,44)(20,35)(21,30)(23,45)(25,31)(26,50)(27,37)(28,38)(34,49)(39,52)(40,59)(41,54)(42,55)(43,57)(46,58)(47,53)(48,60)(51,56)(61,63)(62,64);(4,19,22,49)(5,7)(6,27)(10,34,18,44)(11,24)(12,37)(15,40,21,47)(16,25)(17,43)(20,46,23,48)(26,28)(30,53,36,59)(31,33)(32,57)(35,58,45,60)(38,50)(39,51)(41,61,42,62)(52,56)(54,63,55,64);(3,14)(4,20,22,23)(6,28)(9,29)(10,35,18,45)(12,38)(15,41,21,42)(16,25)(19,46,49,48)(26,27)(30,54,36,55)(31,33)(34,58,44,60)(37,50)(40,61,47,62)(53,63,59,64);(1,3)(2,9)(4,21)(5,25)(6,17)(7,16)(8,14)(10,36)(11,33)(12,32)(13,29)(15,22)(18,30)(19,47)(20,42)(23,41)(24,31)(26,51)(27,43)(28,39)(34,59)(35,55)(37,57)(38,52)(40,49)(44,53)(45,54)(46,62)(48,61)(50,56)(58,64)(60,63);(1,4,6,19,7,23,27,46,8,22,28,49,5,20,26,48)(2,10,12,34,11,35,37,60,13,18,38,44,24,45,50,58)(3,15,17,40,16,42,43,61,14,21,39,47,25,41,51,62)(9,30,32,53,33,54,57,64,29,36,52,59,31,55,56,63)

Polycyclic presentation:

2^10|b^2=h,c^2=h,e^2=g,f^2=j,g^2=i,i^2=j,c^a=ch,d^c=dhj,e^a=ej,e^b=eg,e^c=ef,e^d=ej,f^a=fj,f^b=fj,f^d=fj,g^b=gi,g^c=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249843
maximal normal subgroup of	#72249843
maximal quotient of	#72257033, #72322945

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