#72285124

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
prime radical	#10336	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2199 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)(3,7)(4,8)(5,10)(6,11)(9,18)(12,19)(13,26)(14,21)(15,22)(16,23)(17,24)(20,29)(25,30)(27,31)(28,32)(47,60)(55,62)(58,61)(63,64);(1,3)(2,7)(4,15,17,12)(5,13,18,29)(6,14)(8,22,24,19)(9,20,10,26)(11,21)(16,27)(23,31)(25,32)(28,30)(33,34)(35,43)(36,40)(37,49,51,41)(38,42)(39,46)(44,54)(45,57,50,53)(47,55,60,62)(48,56)(52,59)(58,63,61,64);(1,4)(2,8)(3,12,14,15)(5,16,18,30)(6,17)(7,19,21,22)(9,25,10,23)(11,24)(13,28)(20,31)(26,32)(27,29)(33,35)(34,39)(36,44,48,59)(37,50)(38,46)(40,54,56,52)(41,57)(42,43)(45,51)(47,61,60,58)(49,53)(55,63,62,64);(1,5)(2,9,11,10)(3,13)(4,16,17,30)(6,18)(7,20,21,26)(8,23)(12,27,15,28)(14,29)(19,31)(22,32)(24,25)(33,36,38,48)(34,40)(35,44)(37,47,51,60)(39,52,43,54)(41,55)(42,56)(45,58)(46,59)(49,62)(50,61)(53,63,57,64);(1,5)(2,10,11,9)(3,13)(4,16,17,30)(6,18)(7,20,21,26)(8,23)(12,28,15,27)(14,29)(19,31)(22,32)(24,25)(33,37)(34,41,42,49)(35,45,46,50)(36,47)(38,51)(39,53)(40,55,56,62)(43,57)(44,58,59,61)(48,60)(52,63)(54,64)

Polycyclic presentation:

2^10|b^2=f,c^2=fhij,d^2=ghi,e^2=ghij,c^b=ch,d^a=df,d^b=dh,d^c=dfg,e^a=efhj,e^b=egij,e^c=ef,f^c=fi,f^d=fi,f^e=fij,g^a=gi,g^b=gij,g^c=gj,g^d=gj,g^e=gj,h^b=hi,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599
maximal normal subgroup of	#72285599

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