#72284159

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=f,d^2=hj,b^a=bi,c^b=ch,d^a=dhi,d^b=di,d^c=dg,e^a=efi,e^b=egj,e^d=efj,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72341033, #72341083, …
maximal normal subgroup of	#72284709, #72341033, #72341083, …

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