#72235802

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1304
commutator	C₂³×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#126	C₂³
prime radical	C₂⁶	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

1939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=deij,c^2=j,d^2=i,e^2=ij,f^2=j,b^a=bf,c^a=cghj,c^b=cgij,d^b=dh,d^c=dij,e^a=eg,e^b=eh,e^d=ei,f^a=fhj,f^b=fgj,f^c=fij,f^d=fj,f^e=fi,g^a=gi,g^b=gij,h^a=hj,h^b=hi

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235822, #72235838, #72235849
maximal normal subgroup of	#72235822, #72235838, #72235849

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