#72287366

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

747 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)(3,8)(4,16)(5,11)(6,21)(7,12)(9,14)(10,20)(13,28)(15,24)(17,26)(18,27)(19,31)(22,25)(23,29)(30,32)(33,34,39,42)(35,40,48,59)(36,50,47,41)(37,43,52,62)(38,44,55,56)(45,61,51,57)(46,58,49,60)(53,63,54,64);(1,3,5,14,7,15,6,22)(2,8,11,9,12,24,21,25)(4,17,18,32,19,13,20,29)(10,23,16,26,27,30,31,28)(33,35,37,46,39,48,52,49)(34,40,43,58,42,59,62,60)(36,38,51,53,47,55,45,54)(41,56,61,64,50,44,57,63);(1,3,7,15)(2,9,12,25)(4,17,19,13)(5,14,6,22)(8,11,24,21)(10,26,27,28)(16,30,31,23)(18,32,20,29)(33,36,39,47)(34,41,42,50)(35,38,48,55)(37,51,52,45)(40,56,59,44)(43,61,62,57)(46,53,49,54)(58,64,60,63);(1,4,5,18,7,19,6,20)(2,10,21,31,12,27,11,16)(3,13,22,32,15,17,14,29)(8,23,9,26,24,30,25,28)(34,42)(35,45)(36,49)(37,52)(38,54)(40,57)(41,60)(44,63)(46,47)(48,51)(50,58)(53,55)(56,64)(59,61)

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=cgj,d^a=dg,d^b=df,d^c=dfh,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	#72287387
maximal normal subgroup of	#72287387

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.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);;
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);