#72287742

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	#22858
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#489286	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

896 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,c^2=e,d^2=eghi,e^2=i,f^2=hj,g^2=ij,h^2=j,c^a=cij,c^b=cf,d^a=dg,d^b=dhi,d^c=dfghi,e^b=ej,e^d=ej,f^b=fh,f^c=fhj,f^d=fhj,g^a=gij,g^c=gij,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287776
maximal normal subgroup of	#72287776

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