#72258406

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#115133
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
prime radical	#6437622	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

3993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=j,e^2=j,b^a=be,c^a=cg,c^b=cd,d^a=dh,e^a=ei,e^b=ej,e^c=ehj,f^b=fij,f^c=fj,f^d=fj,g^a=gj,g^b=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258409
maximal normal subgroup of	#72258409
maximal quotient of	#72266222, #72266298

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