#72287794

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	#10580932	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4520 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)(4,18)(5,10)(6,11)(7,12)(8,14)(9,20)(15,22)(16,28)(17,29)(19,26)(21,25)(23,31)(24,32)(27,30)(33,34)(35,46)(36,50)(37,43)(38,44)(39,41)(40,45)(42,51)(47,56)(48,58)(49,57)(52,64)(53,61)(54,62)(55,60)(59,63);(1,3)(2,8)(4,15)(5,16)(6,17)(7,14)(9,22)(10,23)(11,24)(12,13)(18,27)(19,31)(20,30)(21,32)(25,29)(26,28)(33,35,40,39)(34,41,45,46)(36,47,51,54)(37,48,38,49)(42,56,50,62)(43,57,44,58)(52,55,63,53)(59,60,64,61);(3,14)(8,13)(15,30)(16,31)(17,32)(22,27)(23,28)(24,29)(36,51)(42,50)(47,54)(52,63)(53,55)(56,62)(59,64)(60,61);(1,4,6,19,7,20,21,5)(2,9,11,10,12,18,25,26)(3,15,17,31,14,30,32,16)(8,22,24,23,13,27,29,28)(33,36,37,53,39,54,49,63,40,51,38,55,35,47,48,52)(34,42,43,60,41,56,57,64,45,50,44,61,46,62,58,59);(1,5,21,20,7,19,6,4)(2,10,25,9,12,26,11,18)(3,16,32,30,14,31,17,15)(8,23,29,22,13,28,24,27)(35,39)(36,52,51,63)(37,49)(38,48)(41,46)(42,59,50,64)(43,57)(44,58)(47,53,54,55)(56,61,62,60)

Polycyclic presentation:

2^10|b^2=j,d^2=f,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bhi,c^b=chij,d^a=dhi,d^c=dj,e^b=ej,e^d=eg,f^e=fi,g^d=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287809, #72287827, #72318898, …
maximal normal subgroup of	#72287809, #72287827, #72318898, …
maximal quotient of	#72343788

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