#72225522

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#128900
commutator	C₄×C₁₆	C₂×C₄²
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	#491485	C₂²
LCS	G ▷ C₄×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=f,c^2=e,d^2=ghi,f^2=h,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cj,c^b=chij,d^a=dg,d^b=dghj,d^c=dk,e^b=ejk,f^b=fhj,g^a=gi,g^b=gi,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225549
maximal normal subgroup of	#72225549
maximal quotient of	#72225523

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