#72280807

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

5001 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)(4,19,23,40)(5,11)(6,28)(7,12)(8,13)(9,17)(10,22,37,25)(15,44,47,50)(16,43)(18,33)(20,53,56,63)(21,36)(24,52,30,51)(26,60)(27,41)(29,34)(31,55,62,58)(32,49)(35,48,42,46)(38,59,39,57)(45,64)(54,61);(1,3)(2,9)(4,20)(5,16)(6,17,7,29)(8,18)(10,35)(11,32)(12,33,28,14)(13,34)(15,24,47,30)(19,50)(21,45)(22,55)(23,56)(25,58)(26,49,27,54)(31,38,62,39)(36,61)(37,42)(40,44)(41,64,60,43)(46,57,48,59)(51,63,52,53);(1,4,21,22,8,23,5,25)(2,10,36,19,13,37,11,40)(3,15,45,46,18,47,16,48)(6,24,27,57,7,30,26,59)(9,31,61,53,34,62,32,63)(12,38,60,51,28,39,41,52)(14,42,64,44,33,35,43,50)(17,20,54,55,29,56,49,58)

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=dej,d^2=j,f^2=j,b^a=beg,c^a=cd,c^b=cefj,d^c=dj,e^c=ej,f^a=fi,f^b=fi,g^b=gij,g^c=gi,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	#72280854, #72281237, #72282548
maximal normal subgroup of	#72280854, #72281237, #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, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*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, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
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);