#72227010

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#136613
commutator	C₂×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#346	C₂³
prime radical	C₂×#2221	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #33326 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=d,d^2=g,e^2=hi,f^2=j,g^2=j,h^2=ij,i^2=j,b^a=be,c^a=cfj,c^b=cehj,d^a=dij,d^c=dij,e^a=eh,e^b=ehij,e^c=eh,e^d=ej,f^c=fj,g^a=gj,g^c=gj,h^a=hi,h^b=hij,h^c=hi,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227017, #72227147, #72227151, …
maximal normal subgroup of	#72227017, #72227147, #72227151, …

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