#72286535

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#93757
commutator	C₃₂	C₄×C₁₆
Frattini, cosocle	C₂×#7330	C₂²
prime radical	C₂³	#7389
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 68 points:

(1,2,3,4)(6,12,14,32,16,34,15,19,18,36,25,33,46,37,48,54)(7,13,20,39,22,41,56,43)(8,24)(9,26,29,57)(10,31)(17,49,51,35,28,61,60,66,52,68,44,67,63,38,47,62)(21,45,55,40,23,58,50,42)(27,53)(30,65,64,59);(5,6,13,38,59,44,7,19,53,15,45,66,30,52,55,36,31,46,39,49,26,60,56,33,27,14,40,67,9,28,21,37,11,18,41,61,65,51,22,54,24,48,58,62,64,17,50,12,10,16,43,68,57,47,20,32,8,25,42,35,29,63,23,34)

Polycyclic presentation:

2^11|a^2=c,b^2=cdegij,c^2=f,d^2=g,e^2=hj,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bd,c^b=cijk,d^a=dg,d^b=di,d^c=dj,e^a=ej,e^b=ej,e^d=ek,f^b=fjk,f^d=fk,g^a=gi,g^b=gj,g^c=gk,h^b=hk,i^a=ij,i^b=ik,j^a=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286570, #72287062, #72287177, …
maximal normal subgroup of	#72286570, #72287062, #72287177, …

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