#72285451

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	#21503
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#19603	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

872 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ehi,c^2=gh,d^2=eij,e^2=h,f^2=hi,g^2=i,i^2=j,b^a=bg,c^a=cghj,c^b=cf,d^a=dij,d^b=dhj,d^c=dhij,e^b=ej,e^c=ej,f^a=fi,f^b=fhi,f^c=fh,f^d=fj,g^a=gi,g^b=gi,g^d=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285671, #72287651
maximal normal subgroup of	#72285671, #72287651

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