#72307218

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346542
commutator, Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10583462	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3682 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:

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

Polycyclic presentation:

2^10|a^2=hj,b^2=h,c^2=efj,d^2=h,e^2=j,f^2=i,g^2=j,i^2=j,c^a=ce,c^b=cf,d^a=dh,d^c=degj,e^b=eh,e^c=eh,e^d=eh,f^a=fhj,f^b=fi,f^c=fh,f^d=fh,f^e=fj,g^a=gh,g^b=gj,g^c=ghj,g^d=gh,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307300, #72307752, #72308079
maximal normal subgroup of	#72307300, #72307752, #72308079

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