#72317522

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9497
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#119494
prime radical	#62811	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2634 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=i,f^2=j,b^a=be,c^a=cfg,c^b=cgj,d^b=dh,d^c=di,e^a=eh,e^c=eij,g^a=gij,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317535, #72317598, #72317779
maximal normal subgroup of	#72317535, #72317598, #72317779

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