#72287470

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#3444	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2141 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)(3,7)(4,10)(5,11)(6,9)(8,12)(13,19)(14,20)(15,21)(16,25)(17,23)(18,24)(22,26)(27,30)(28,31)(29,32)(33,34)(35,40)(36,48)(37,43)(38,44)(39,42)(41,50)(45,56)(46,54)(47,55)(49,57)(51,64)(52,61)(53,60)(58,62)(59,63);(1,3)(2,7)(4,15)(5,14)(6,12)(8,9)(10,21)(11,20)(13,17)(16,28)(18,26)(19,23)(22,24)(25,31)(27,29)(30,32)(33,34)(35,40)(36,48)(37,43)(38,44)(39,42)(41,50)(45,56)(46,54)(47,55)(49,57)(51,64)(52,61)(53,60)(58,62)(59,63);(1,3)(2,8)(4,13)(5,14)(6,12)(7,9)(10,21)(11,22)(15,17)(16,27)(18,26)(19,23)(20,24)(25,31)(28,29)(30,32)(33,35,37,36,38,47,52,51,39,46,53,50,45,62,49,63)(34,40,43,48,44,55,61,64,42,54,60,41,56,58,57,59);(2,9)(3,12)(10,23)(11,24)(13,15)(14,26)(25,32)(27,28)(33,36,49,35,45,63,53,62,39,50,52,46,38,51,37,47)(34,41,57,54,56,64,60,55,42,48,61,40,44,59,43,58);(1,4,5,16,6,17,18,29)(2,10,11,25,9,23,24,32)(3,13,14,27,12,15,26,28)(7,19,20,30,8,21,22,31)(34,42)(35,36,46,50)(37,52)(38,45)(40,41,54,48)(43,57)(47,63,62,51)(49,53)(55,64,58,59)(60,61)

Polycyclic presentation:

2^10|c^2=g,d^2=gij,e^2=f,f^2=h,g^2=i,i^2=j,c^a=ch,d^a=dhj,d^b=dhj,d^c=dhj,e^a=ej,e^b=ehj,e^c=egij,e^d=egj,f^c=fj,f^d=fj,g^e=gi,i^e=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287517, #72287605, #72287792, …
maximal normal subgroup of	#72287517, #72287605, #72287792, …

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