#72287387

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	D₄×#442
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72287259	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #184045 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=j,b^2=h,c^2=f,d^2=hj,e^2=fj,f^2=j,g^2=i,h^2=jk,i^2=k,b^a=bh,c^b=chjk,d^a=dhjk,d^b=djk,d^c=dg,e^a=ejk,e^c=ej,e^d=eijk,f^d=fk,g^a=gjk,g^c=gi,g^d=gij,g^e=gk,h^a=hjk,h^c=hjk,i^c=ik,i^d=ik

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

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