#72287783

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	#7623516	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2192 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)(4,17)(5,10)(6,11)(7,12)(9,19)(13,21)(14,30)(15,23)(16,24)(18,26)(20,25)(22,27)(28,31)(29,32)(33,34)(35,41)(36,51)(37,43)(38,44)(39,45)(40,46)(42,52)(47,56)(48,62)(49,57)(50,58)(53,60)(54,61)(55,63)(59,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,13)(9,22)(10,23)(11,24)(12,21)(17,30)(18,28)(19,27)(20,29)(25,32)(26,31)(33,35,38,49,39,50,55,36,40,47,37,53,48,64,54,52)(34,41,44,57,45,58,63,51,46,56,43,60,62,59,61,42);(3,13)(8,21)(14,27)(15,28)(16,29)(22,30)(23,31)(24,32)(35,47)(36,52)(41,56)(42,51)(49,53)(50,64)(57,60)(58,59);(1,4,6,18,7,19,20,5)(2,9,11,10,12,17,25,26)(3,14,16,28,13,27,29,15)(8,22,24,23,21,30,32,31)(33,36,37,47,39,53,54,64,40,52,38,35,48,49,55,50)(34,42,43,41,45,57,61,58,46,51,44,56,62,60,63,59);(1,5,20,19,7,18,6,4)(2,10,25,9,12,26,11,17)(3,15,29,27,13,28,16,14)(8,23,32,22,21,31,24,30)(35,49,47,53)(36,50,52,64)(37,54)(38,55)(39,48)(41,57,56,60)(42,59,51,58)(43,61)(44,63)(45,62)

Polycyclic presentation:

2^10|b^2=g,d^2=f,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,c^b=chi,d^a=dhi,d^b=dij,d^c=dj,e^b=egij,e^d=eg,f^b=fj,f^e=fi,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	#72287809, #72287827, #72318539, …
maximal normal subgroup of	#72287809, #72287827, #72318539, …
maximal quotient of	#72343809

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, 4, F.6);;
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, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
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);