#72225036

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#11663
commutator	C₂×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#346	C₂³
socle	C₂	#129503
prime radical	C₂×#3414	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×C₄×M₃₂ ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1155 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:

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

Polycyclic presentation:

2^10|a^2=d,d^2=g,f^2=hj,g^2=ij,h^2=i,i^2=j,b^a=bf,c^a=ce,d^b=dh,d^c=dj,e^a=ej,e^b=ej,f^a=fi,f^b=fhi,f^c=fj,f^d=fj,g^b=gi,h^a=hj,h^b=hij,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225046, #72225065, #72225073
maximal normal subgroup of	#72225046, #72225065, #72225073

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