#72287701

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#60673	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=g,d^2=f,e^2=fhi,f^2=h,g^2=i,i^2=j,c^a=ch,d^a=dgi,d^b=dhj,d^c=dj,e^a=eij,e^d=eij,f^a=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287726, #72287751, #72320823
maximal normal subgroup of	#72287726, #72287751, #72320823

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