#72307176

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	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8350
prime radical	#10577631	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^10|a^2=j,d^2=f,e^2=ij,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bhij,c^a=cj,c^b=cj,d^a=de,d^b=degj,e^a=ei,e^b=eh,f^a=fij,f^b=fi,g^a=ghij,g^b=ghij,g^d=gj,g^e=gj,h^a=hj,h^b=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307207, #72307300, #72307386
maximal normal subgroup of	#72307207, #72307300, #72307386

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