#72319954

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2606478
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

5616 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)(4,21,25,37)(5,12)(6,14)(7,13)(8,9)(10,19)(11,22,32,29)(16,34,46,52)(17,44)(18,31)(20,36)(23,54,50,38)(24,53,56,41)(26,58,57,39)(27,43)(28,33)(30,42)(35,51)(40,48)(45,60,59,64)(47,63,55,61)(49,62);(2,9)(4,22)(6,7)(11,37)(12,33)(15,36)(16,23)(18,19)(21,32)(24,26)(25,29)(27,30)(34,54)(38,52)(39,41)(44,62)(45,47)(46,50)(48,51)(53,58)(55,59)(56,57)(60,63)(61,64);(1,3,8,20)(2,10,9,31)(4,23,25,50)(5,17,28,49)(6,19,7,18)(11,38,32,54)(12,40,33,35)(13,15,14,36)(16,29,46,22)(21,52,37,34)(24,55,56,47)(26,59,57,45)(27,51,30,48)(39,61,58,63)(41,60,53,64)(42,62,43,44);(1,4)(2,11)(3,16)(5,26)(6,29)(7,22)(8,25)(9,32)(10,34)(12,41)(13,21)(14,37)(15,38)(17,47)(18,50)(19,23)(20,46)(24,27)(28,57)(30,56)(31,52)(33,53)(35,61)(36,54)(39,42)(40,63)(43,58)(44,64)(45,48)(49,55)(51,59)(60,62);(1,5)(2,12)(3,17)(4,24)(6,27)(7,30)(8,28)(9,33)(10,35)(11,39)(13,42)(14,43)(15,44)(16,45)(18,48)(19,51)(20,49)(21,53)(22,26)(23,47)(25,56)(29,57)(31,40)(32,58)(34,60)(36,62)(37,41)(38,61)(46,59)(50,55)(52,64)(54,63)

Polycyclic presentation:

2^10|a^2=g,c^2=j,f^2=j,h^2=j,b^a=bgj,c^a=ch,c^b=cgi,d^a=dhij,d^b=dhij,d^c=df,e^c=ei,e^d=egh,f^a=fj,f^b=fj,f^c=fj,f^d=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	#72320148
maximal normal subgroup of	#72320148

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