#72280796

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10995
commutator	C₂×#41	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#123590
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

5041 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)(3,14,17,43)(4,20,24,39)(5,10)(6,12)(7,13)(8,19,33,18)(9,23,36,28)(11,30)(15,46)(16,45,27,61)(21,54)(22,35)(25,60,26,53)(29,41)(31,55)(32,52,38,51)(34,47)(37,59,42,58)(40,56)(44,50)(48,62)(49,64)(57,63);(1,3)(2,8)(4,21)(5,27)(6,18,30,19)(7,17)(9,34)(10,38)(11,14,12,43)(13,33)(15,26,49,25)(16,22)(20,46)(23,57)(24,48)(28,55)(29,52,56,51)(31,42,63,37)(32,35)(36,44)(39,64)(40,61,41,45)(47,58,50,59)(53,54,60,62);(1,4,22,23,7,24,5,28)(2,9,35,20,13,36,10,39)(3,15,16,50,17,49,27,47)(6,26,56,59,30,25,29,58)(8,31,32,54,33,63,38,62)(11,42,41,60,12,37,40,53)(14,44,45,64,43,34,61,46)(18,48,51,55,19,21,52,57)

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=dej,d^2=j,g^2=j,b^a=begj,c^a=cd,c^b=cef,d^b=dj,d^c=dj,e^b=ej,e^c=ej,f^a=fi,f^b=fi,f^c=fj,g^b=gij,g^c=gij,g^f=gj,h^a=hij,h^c=hij,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280876, #72281160, #72282548
maximal normal subgroup of	#72280876, #72281160, #72282548

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