#72303296

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#11087
commutator	C₂²×D₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#124038
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3875 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)(3,8)(4,17,20,9)(5,10)(6,12)(7,11)(13,39,42,27)(14,28)(15,38)(16,29)(18,31,48,44)(19,45,50,32)(21,46,40,33)(22,34)(23,36)(24,35)(25,30)(26,37)(41,60,52,55)(43,61)(47,57,49,58)(51,64,63,59)(53,56)(54,62);(1,3,7,16)(2,8,11,29)(4,18)(5,22,24,14)(6,25)(9,31)(10,34,35,28)(12,38)(13,40,42,21)(15,26)(17,44)(19,49)(20,48)(23,43)(27,46,39,33)(30,37)(32,58)(36,56)(41,51,52,63)(45,57)(47,50)(53,54)(55,59,60,64)(61,62);(1,4,5,19,7,20,24,50)(2,9,10,32,11,17,35,45)(3,13,14,41,16,42,22,52)(6,21,23,51,26,40,54,63)(8,27,28,55,29,39,34,60)(12,33,36,59,37,46,62,64)(15,18,43,47,25,48,53,49)(30,31,56,57,38,44,61,58)

Polycyclic presentation:

2^10|a^2=f,b^2=hj,c^2=d,d^2=j,g^2=j,b^a=bh,c^a=ce,c^b=cg,d^b=dj,e^a=ej,f^c=fj,g^a=gi,g^b=gi,h^c=hij,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303336, #72303460, #72303500
maximal normal subgroup of	#72303336, #72303460, #72303500

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