#72281697

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#121601
commutator	C₂×#131	C₂²×C₄
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

1200 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=g,c^2=ej,e^2=i,f^2=i,h^2=j,i^2=j,b^a=bg,c^a=ce,c^b=cf,d^c=dj,e^a=ei,e^b=ei,f^a=fh,f^b=fh,g^c=ghi,g^f=gj,h^a=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281861
maximal normal subgroup of	#72281861

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