#72287532

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23029
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#490656	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=i,d^2=gi,e^2=i,f^2=h,g^2=ij,h^2=j,c^a=cg,c^b=cgij,d^a=dg,d^b=df,d^c=dfhi,e^d=ej,f^a=fij,f^b=fh,f^c=fhj,f^d=fhi,g^a=gij,g^b=gij,g^c=gij,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287545
maximal normal subgroup of	#72287545

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