#72226831

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#6453969
commutator, Frattini, cosocle	C₂³×C₈	C₂⁵
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #6369990 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

13910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^11|a^2=h,c^2=f,d^2=g,e^2=g,h^2=j,j^2=k,b^a=bi,c^b=cg,d^a=dhjk,d^b=dg,d^c=df,e^a=ejk,g^a=gk,h^b=hk,h^d=hj,h^e=hk,i^a=ik,i^c=ik,j^d=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72226832

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