#72307245

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21408
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#354203
prime radical	#60450	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2637 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fgij,c^2=efhij,d^2=ij,e^2=ij,f^2=i,g^2=i,h^2=j,i^2=j,b^a=bf,c^a=ci,d^a=dj,d^b=dfghij,e^b=ei,e^c=ej,f^a=fj,f^b=fi,f^c=fj,f^d=fj,g^b=gi,g^c=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72320043, #72320817
maximal normal subgroup of	#72307301, #72320043, #72320817

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