#72320016

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#385
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#354413
prime radical	C₂²×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2127 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=e,c^2=efij,d^2=j,e^2=i,f^2=i,i^2=j,b^a=bg,c^a=ci,c^b=ci,d^a=dh,e^c=ej,f^a=fj,f^b=fj,f^c=fj,g^d=gj,h^b=hj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320043, #72320350, #72343973
maximal normal subgroup of	#72320043, #72320350, #72343973

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