#72287743

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	#7623520	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2268 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)(3,14)(4,10)(5,11)(6,12)(7,9)(8,13)(15,28)(16,24)(17,23)(18,25)(19,26)(20,22)(21,27)(29,32)(30,31)(34,41)(42,51)(43,57)(44,53)(45,58)(59,62)(60,61)(63,64);(1,3,8,7)(2,9,13,14)(4,15,21,20)(5,17,6,16)(10,22,27,28)(11,24,12,23)(18,30,19,29)(25,32,26,31)(33,34,38,44,39,45,52,59,40,41,37,53,54,58,46,62)(35,43,48,64,49,42,56,61,50,57,55,63,36,51,47,60);(1,4,7,20,8,21,3,15)(2,10,9,22,13,27,14,28)(5,19,17,29,6,18,16,30)(11,26,23,32,12,25,24,31)(33,35,37,48,39,49,46,56,40,50,38,55,54,36,52,47)(34,42,44,60,45,57,59,64,41,51,53,61,58,43,62,63);(1,5,3,16,8,6,7,17)(2,11,14,24,13,12,9,23)(4,18,15,29,21,19,20,30)(10,25,28,32,27,26,22,31)(33,36,37,47,39,35,46,48,40,49,38,56,54,50,52,55)(34,43,44,63,45,42,59,60,41,57,53,64,58,51,62,61);(1,6,7,16,8,5,3,17)(2,12,9,24,13,11,14,23)(4,19,20,29,21,18,15,30)(10,26,22,32,27,25,28,31)(34,44,41,53)(35,47,50,56)(36,48,49,55)(37,46)(38,52)(39,54)(42,51)(45,62,58,59)(60,63)(61,64)

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=fhij,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bhi,c^b=cij,d^b=dhj,d^c=dhi,e^b=egij,e^c=eg,e^d=egj,f^b=fj,f^e=fi,g^c=gj,g^d=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287790, #72287809, #72287827, …
maximal normal subgroup of	#72287790, #72287809, #72287827, …
maximal quotient of	#72343805

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