#72285671

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#21503
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72285618	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #180817 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4970 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=fj,c^2=h,d^2=ik,e^2=fj,f^2=i,g^2=ij,h^2=j,j^2=k,b^a=bh,c^a=ch,c^b=cg,d^c=di,e^a=ej,e^c=eijk,f^b=fk,f^c=fk,g^a=gj,g^b=gij,g^c=gi,g^e=gk,h^a=hj,h^b=hj,h^e=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

automorphism group of

#7453

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.6*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.9*F.11);;
gap> SetPower(coll, 5, F.6*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);