#72342809

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10336
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#111458
prime radical	#6437628	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=g,c^2=i,d^2=j,b^a=bf,c^a=ce,c^b=cgij,d^b=dh,d^c=di,e^a=ei,e^b=ehij,f^a=fh,f^b=fh,f^c=fij,f^e=fj,g^a=gh,g^c=gj,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342816, #72342829, #72342838
maximal normal subgroup of	#72342816, #72342829, #72342838

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