#72307828

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	#60968
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	#6453737
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5150 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:

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

Polycyclic presentation:

2^10|b^2=gj,c^2=gj,e^2=f,f^2=i,i^2=j,b^a=bg,d^b=dg,d^c=dj,e^a=eh,e^b=ef,e^c=ei,e^d=egj,f^b=fi,f^c=fj,f^d=fj,g^e=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308106
maximal normal subgroup of	#72308106

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