#72285176

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

2259 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)(5,17)(6,10)(9,20)(11,21)(12,22)(13,23)(14,24)(15,26)(16,27)(18,30)(19,25)(28,31)(29,32)(36,46)(37,48)(40,55)(41,49)(43,58)(44,50)(47,51)(52,62)(53,60)(56,57)(59,61)(63,64);(1,3)(2,7)(4,14,16,11)(5,18)(6,13)(8,24,27,21)(9,22)(10,23)(12,20)(15,28,19,29)(17,30)(25,32,26,31)(33,34)(35,39)(36,40,46,55)(37,49,48,41)(38,42)(43,52,58,62)(44,60,50,53)(45,54)(47,56)(51,57)(59,63)(61,64);(1,4)(2,8)(3,11)(5,19)(6,16)(7,21)(9,26)(10,27)(12,28)(13,14)(15,20)(17,25)(18,29)(22,31)(23,24)(30,32)(33,35,38,45)(34,39,42,54)(36,43,46,58)(37,50,48,44)(40,52,55,62)(41,53,49,60)(47,59,51,61)(56,64,57,63);(1,5,6,20)(2,9)(3,12)(4,15)(7,22,23,30)(8,25,27,26)(10,17)(11,28,14,29)(13,18)(16,19)(21,31)(24,32)(33,36)(34,40,42,55)(35,43)(37,51,48,47)(38,46)(39,52,54,62)(41,57)(44,61,50,59)(45,58)(49,56)(53,64)(60,63);(1,5)(2,9,10,17)(3,12,13,18)(4,15)(6,20)(7,22)(8,26,27,25)(11,28,14,29)(16,19)(21,32)(23,30)(24,31)(33,37)(34,41,42,49)(35,44)(36,47,46,51)(38,48)(39,53,54,60)(40,56)(43,59,58,61)(45,50)(52,63)(55,57)(62,64)

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=ghi,e^2=g,c^b=cf,d^a=dfj,d^b=dfg,d^c=df,e^a=efhj,e^b=efgj,e^c=egij,e^d=ehij,f^b=fi,f^d=fi,f^e=fij,g^a=gi,g^b=gij,g^d=gj,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

61 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.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
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, 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, 4, F.7*F.10);;
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);