#72286548

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#12759
commutator	#124	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	#10576177	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5896 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)(3,8)(4,17,20,49)(5,22)(6,26)(7,10)(9,19,36,42)(11,24)(12,29)(13,41,45,35)(14,33)(15,32)(16,34)(18,30,44,61)(21,50,28,51)(23,38)(25,39)(27,31)(37,54,40,55)(43,63,46,64)(47,62)(48,53,60,52)(56,58)(57,59);(1,3)(2,8)(4,18,21,53)(5,23,24,58)(6,27,29,14)(7,16)(9,35,37,63)(10,34)(11,38,22,56)(12,31,26,33)(13,42,46,55)(15,25)(17,48,50,61)(19,43,54,45)(20,44,28,52)(30,49,60,51)(32,39)(36,41,40,64)(47,59)(57,62);(1,4)(2,9)(3,13)(5,19)(6,28)(7,20)(8,30)(10,36)(11,17)(12,40)(14,46)(15,18)(16,45)(21,29)(22,49)(23,53)(24,42)(25,54)(26,37)(27,43)(31,48)(32,35)(33,60)(34,61)(38,63)(39,50)(41,62)(44,47)(51,57)(52,58)(55,59)(56,64);(2,10)(3,14,16,27)(8,31,34,33)(9,36)(11,22)(12,26)(13,43,45,46)(15,23,47,58)(17,49)(18,52,44,53)(30,60,61,48)(32,56,62,38)(35,63,41,64)(37,40)(39,57)(50,51)

Polycyclic presentation:

2^10|a^2=f,b^2=g,d^2=ij,e^2=ij,h^2=j,b^a=bfg,c^a=ce,c^b=cej,d^a=dj,d^b=dhij,d^c=dij,e^a=ei,e^b=eh,e^c=eij,f^c=fj,g^c=ghi,g^d=gij,g^e=gi,h^a=hi,h^b=hij,h^c=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286575, #72286867, #72287026
maximal normal subgroup of	#72286575, #72286867, #72287026

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