#72318890

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2606862
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#6437578	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5392 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)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,19)(10,26)(16,46)(17,45)(18,32)(20,35)(22,39)(23,36)(24,53)(25,38)(27,56)(28,58)(29,43)(30,42)(31,41)(33,48)(34,52)(37,55)(40,54)(44,49)(47,63)(50,61)(51,59)(57,62)(60,64);(4,22)(6,7)(9,32)(10,21)(12,13)(16,23)(18,19)(24,28)(25,26)(29,31)(33,36)(34,59)(37,53)(38,39)(40,58)(41,43)(44,46)(47,50)(48,49)(51,52)(54,55)(57,64)(60,61)(62,63);(1,3,8,20)(2,9,14,32)(4,23,25,49)(5,27,30,17)(6,19,7,18)(10,36,39,44)(11,34,42,59)(12,15,13,35)(16,26,48,22)(21,46,38,33)(24,50,54,57)(28,47,55,64)(29,51,31,52)(37,62,58,61)(40,63,53,60)(41,45,43,56);(1,4,7,26,8,25,6,22)(2,10,12,38,14,39,13,21)(3,16,19,49,20,48,18,23)(5,28,31,24,30,55,29,54)(9,33,35,36,32,46,15,44)(11,40,41,58,42,53,43,37)(17,50,52,47,27,57,51,64)(34,61,56,63,59,62,45,60);(1,5)(2,11)(3,17)(4,24)(6,29)(7,31)(8,30)(9,34)(10,37)(12,41)(13,43)(14,42)(15,45)(16,47)(18,51)(19,52)(20,27)(21,53)(22,28)(23,50)(25,54)(26,55)(32,59)(33,60)(35,56)(36,61)(38,40)(39,58)(44,62)(46,63)(48,64)(49,57)

Polycyclic presentation:

2^10|c^2=j,d^2=fgij,f^2=j,h^2=j,b^a=bg,c^a=ch,c^b=cgi,d^a=dghj,d^b=dghij,d^c=df,e^c=eij,e^d=eghj,f^a=fj,f^b=fj,f^c=fj,g^d=gj,h^a=hj,h^b=hj,h^c=hj,h^e=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319020
maximal normal subgroup of	#72319020

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