#72318655

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	#7623523	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2342 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)(4,9)(5,19)(6,11)(7,12)(10,20)(13,22)(14,24)(15,25)(16,26)(17,23)(18,28)(21,29)(27,31)(30,32)(33,34)(35,40)(36,50)(37,42)(38,43)(39,44)(41,52)(45,55)(46,59)(47,57)(48,58)(49,60)(51,63)(53,61)(54,62)(56,64);(1,3,6,14,7,15,21,31)(2,8,11,24,12,25,29,27)(4,16,18,20,17,13,30,5)(9,26,28,10,23,22,32,19)(33,35,37,48,39,47,53,49)(34,40,42,58,44,57,61,60)(36,46,51,45,52,64,38,54)(41,56,43,62,50,59,63,55);(1,4)(2,9)(3,13)(5,14)(6,18)(7,17)(8,22)(10,27)(11,28)(12,23)(15,16)(19,24)(20,31)(21,30)(25,26)(29,32)(35,45)(36,51)(37,53)(38,52)(40,55)(41,43)(42,61)(46,48)(47,54)(49,64)(50,63)(56,60)(57,62)(58,59);(1,5,6,16,7,20,21,13)(2,10,11,22,12,19,29,26)(3,4,14,18,15,17,31,30)(8,23,24,32,25,9,27,28)(33,36,37,51,39,52,53,38)(34,41,42,43,44,50,61,63)(35,46,48,45,47,64,49,54)(40,56,58,62,57,59,60,55);(4,17)(5,20)(9,23)(10,19)(13,16)(18,30)(22,26)(28,32)(35,47)(40,57)(45,54)(46,64)(48,49)(55,62)(56,59)(58,60)

Polycyclic presentation:

2^10|b^2=f,d^2=f,f^2=h,g^2=i,i^2=j,c^b=cg,d^a=dh,d^c=di,e^b=ej,e^c=ehj,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318707, #72318999, #72319316
maximal normal subgroup of	#72318707, #72318999, #72319316

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