#72283809

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

2444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=i,c^2=f,d^2=hi,b^a=bij,c^a=ci,c^b=ch,d^a=dh,d^b=dij,d^c=dg,e^a=efi,e^b=eg,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
maximal normal subgroup of	#72284709

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