#72287019

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1368
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂²×#346	C₂³
prime radical	C₂⁵	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×C₄×C₈ ◁ C₂³×#346 ◁ G
LECS	G ▷ C₂²×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,4);(1,2,3,4)(5,6)(7,8)(9,10,15,16,30,35,28,34,27,17,33,40,26,38,23,37,14,20,29,32,25,39,24,36,13,21,22,31,12,19,11,18);(2,4)(6,8)(10,16,17,31,20,32,21,40)(11,22,23,33)(12,25)(13,27)(15,28,29,24)(18,19,36,35,37,38,34,39)(26,30)

Polycyclic presentation:

2^11|b^2=efijk,c^2=d,d^2=h,f^2=i,g^2=j,i^2=j,j^2=k,b^a=be,c^b=cefgijk,d^b=dj,f^c=fi,f^d=fk,g^c=gj,h^b=hk,i^c=ij,j^c=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287029, #72287081, #72287618
maximal normal subgroup of	#72287029, #72287081, #72287618

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