#72245124

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2380
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	#6357949	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2415 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=gi,e^2=h,g^2=i,h^2=j,d^a=dh,d^b=dij,d^c=dg,e^a=eghij,e^b=ei,e^c=ehj,f^d=fi,f^e=fj,g^a=gj,g^c=gi,h^a=hij,h^c=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72245125, #72245429
maximal subgroup of	#72245125, #72245425
maximal normal subgroup of	#72245125, #72245425

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