#72341232

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2392 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=fij,c^2=g,d^2=j,e^2=gh,b^a=bf,c^a=ch,c^b=chij,d^a=dj,d^b=dfi,d^c=dfhij,e^a=ef,e^b=egi,e^c=efghj,e^d=eh,f^c=fi,f^e=fij,g^a=gij,g^b=gij,g^d=gj,g^e=gj,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	#72341236
maximal normal subgroup of	#72341236

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