#72307252

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#23960
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#364486
prime radical	C₂×#24285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3893 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=e,d^2=i,e^2=i,g^2=i,i^2=j,b^a=bf,c^a=ch,c^b=cg,d^a=dfhj,d^b=dfhi,d^c=dfhj,e^b=ei,e^d=ej,f^c=fj,f^d=fj,g^b=gi,h^b=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72343946, #72343978
maximal normal subgroup of	#72307301, #72343946, #72343978

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