#72283912

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	#245273	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2261 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=ij,c^2=f,d^2=hi,b^a=bj,c^a=ci,c^b=ch,d^a=dh,d^b=di,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

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72285599
maximal normal subgroup of	#72284709, #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, 1, F.9);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*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);;
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);