#72284658

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

2362 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fgh,e^2=gh,b^a=bg,c^a=ch,c^b=cij,d^a=dhij,d^b=dfghi,d^c=dfgh,e^a=efgj,e^b=efghi,e^c=eg,e^d=ehij,f^b=fi,f^c=fj,f^d=fj,f^e=fi,g^a=gi,g^c=gi,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hij,h^e=hj

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284848
maximal normal subgroup of	#72284848

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