#72341248

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237721
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ #21746 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

5878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,13)(4,17)(5,20)(6,12)(7,9)(8,16)(10,19)(11,22)(14,23)(15,25)(18,27)(21,26)(24,29)(28,31)(30,32);(1,3,7,16)(2,8,9,13)(4,18,19,30)(5,14,22,29)(6,15,21,31)(10,27,17,32)(11,24,20,23)(12,25,26,28);(2,9)(3,14)(8,23)(10,17)(11,20)(12,26)(13,24)(15,18)(16,29)(25,32)(27,28)(30,31);(1,4)(2,10)(5,21)(6,22)(7,19)(9,17)(11,26)(12,20)(14,29)(23,24)

Polycyclic presentation:

2^11|b^2=k,f^2=hijk,b^a=bh,c^a=ck,c^b=ce,d^a=dj,d^b=defhijk,d^c=dgj,e^a=ej,e^d=eg,f^a=fjk,f^b=fh,f^c=fgk,f^d=fgh,f^e=fij,g^b=gi,g^f=gjk,h^c=hjk,h^e=hk,h^f=hk,i^d=ij,i^f=ik,j^b=jk

Conjugacy classes & character table

59 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, 2, F.11);;
gap> SetPower(coll, 6, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);