#72319300

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10580912	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4671 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,30)(16,27)(17,23)(18,26)(19,24)(20,28)(21,25)(22,29)(31,32)(35,44)(40,56)(45,62)(46,47)(48,55)(49,53)(57,61)(59,63);(1,3,6,16,7,15,18,21)(2,8,11,25,12,13,26,27)(4,17,19,31,20,5,14,22)(9,10,24,29,28,23,30,32)(33,34,36,42,38,43,51,60,39,41,37,52,50,58,54,64)(35,45,46,57,48,59,49,56,44,62,47,61,55,63,53,40);(1,4,18,14,7,20,6,19)(2,9,26,30,12,28,11,24)(3,5,21,31,15,17,16,22)(8,23,27,29,13,10,25,32)(33,35,39,44)(34,40)(36,49,37,53)(38,55,50,48)(41,56)(42,59)(43,61)(45,60)(46,54,47,51)(52,63)(57,58)(62,64);(1,5)(2,10)(3,14)(4,16)(6,22)(7,17)(8,24)(9,27)(11,29)(12,23)(13,30)(15,19)(18,31)(20,21)(25,28)(26,32)(35,44)(40,56)(45,62)(46,47)(48,55)(49,53)(57,61)(59,63);(3,15)(5,17)(8,13)(10,23)(16,21)(22,31)(25,27)(29,32)(34,41)(40,56)(42,52)(43,58)(45,62)(57,61)(59,63)(60,64)

Polycyclic presentation:

2^10|b^2=f,c^2=fgij,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bhij,c^a=cj,c^b=cg,d^c=dhi,e^b=ehi,e^d=ehij,f^c=fi,g^b=gj,g^c=gi,h^c=hj,i^c=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319316, #72319381, #72319749
maximal normal subgroup of	#72319316, #72319381, #72319749

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