#72307185

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21538
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#359121
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2063 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=efi,d^2=f,e^2=ij,f^2=ij,g^2=i,i^2=j,b^a=bg,c^a=ci,d^a=dfgij,d^b=dfhij,d^c=dij,e^b=ei,e^c=ej,e^d=ej,f^a=fj,f^b=fi,f^c=fj,g^a=gj,g^b=gi,g^c=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72320043, #72320350
maximal normal subgroup of	#72307299, #72320043, #72320350

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