#72286809

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14357
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#6348445	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

2357 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=j,d^2=hij,e^2=j,f^2=i,g^2=ij,h^2=j,b^a=bf,c^b=cfgij,d^a=dghj,d^b=dghij,d^c=dg,e^a=ei,e^d=ej,f^a=fi,f^c=fi,f^d=fj,g^b=gj,g^c=gij,h^a=hi,h^b=hi,h^c=hij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286851
maximal normal subgroup of	#72286851

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