#72307258

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	#24248
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#364572
prime radical	#7622812	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3778 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=e,d^2=efgi,e^2=j,g^2=i,i^2=j,b^a=bf,c^a=cfh,c^b=cfhij,d^a=dhj,d^b=dg,d^c=dfhi,e^b=ej,e^d=ej,f^c=fj,f^d=fj,g^b=gi,g^c=gj,h^b=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72343936, #72343971
maximal normal subgroup of	#72307301, #72343936, #72343971

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