#72225008

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#500908
commutator, Frattini, cosocle	C₂²×C₁₆	C₂⁴
prime radical	#33981	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ #33474 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

950 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=g,d^2=g,e^2=j,f^2=hij,h^2=ij,i^2=j,b^a=bfij,c^a=cij,d^a=de,d^b=dg,d^c=dg,e^a=ej,e^b=ej,e^c=ej,f^b=fh,f^c=fj,g^a=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225019
maximal normal subgroup of	#72225019

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