#72307111

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	C₄	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#24089
prime radical	#10576601	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4741 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)(3,10)(4,19,22,48)(5,14)(6,27)(7,12)(8,9)(11,23,31,54)(13,29)(15,30)(16,42,21,50)(17,39)(18,47)(20,49,24,51)(25,34)(26,32)(28,40)(33,45,37,53)(35,46)(36,55,38,52)(41,60)(43,61,44,63)(56,57)(58,64,59,62);(2,9)(10,30)(11,31)(12,14)(13,32)(19,48)(27,40)(33,37)(34,39)(35,57)(36,38)(42,50)(47,60)(49,51)(58,59)(61,63);(3,15)(4,20,22,24)(6,28)(10,30)(11,36,31,38)(13,32)(16,43,21,44)(17,25)(19,49,48,51)(23,55,54,52)(26,29)(27,40)(33,58,37,59)(34,39)(42,61,50,63)(45,64,53,62);(1,3)(2,10)(4,21)(5,25)(6,18)(7,17)(8,15)(9,30)(11,37)(12,39)(13,35)(14,34)(16,22)(19,50)(20,44)(23,53)(24,43)(26,56)(27,47)(28,41)(29,46)(31,33)(32,57)(36,59)(38,58)(40,60)(42,48)(45,54)(49,63)(51,61)(52,64)(55,62);(1,4,6,23,7,24,29,55,8,22,28,54,5,20,26,52)(2,11,13,19,14,38,27,49,9,31,32,48,12,36,40,51)(3,16,18,45,17,44,46,64,15,21,41,53,25,43,56,62)(10,33,35,42,34,59,47,61,30,37,57,50,39,58,60,63)

Polycyclic presentation:

2^10|a^2=g,c^2=g,e^2=h,f^2=j,h^2=i,i^2=j,b^a=bj,d^c=dgj,e^a=eh,e^c=ef,e^d=ej,f^a=fj,f^d=fj,g^e=gj,h^a=hi,h^c=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307129, #72307386, #72308212, …
maximal normal subgroup of	#72307129, #72307386, #72308212, …
maximal quotient of	C₂×#72307111

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