#72303330

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10960
commutator	C₂⁴	C₄³
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#123452
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#3428 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=h,c^2=f,d^2=j,b^a=bg,c^a=ce,d^b=dj,d^c=dj,e^a=ej,e^c=ej,f^a=fj,g^a=gj,g^b=gi,h^a=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303336, #72303494, #72303509
maximal normal subgroup of	#72303336, #72303494, #72303509

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