#72306891

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#496225
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	#6437454	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ C₂×#3500 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=i,g^2=i,h^2=j,b^a=bf,c^a=cefgij,d^a=defghij,d^b=defghij,e^b=eh,e^c=eij,f^a=fhj,f^b=fhi,f^c=fij,f^d=fj,f^e=fij,g^a=ghij,g^d=gj,g^e=gi,g^f=gi,h^a=hi,h^b=hij,h^e=hj,h^f=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306893, #72307002, #72318012, …
maximal normal subgroup of	#72306893, #72307002, #72318012, …

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