#72258147

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#442
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂⁴
socle	C₂³	C₂×#1580
prime radical	C₂²×D₁₆	C₂×C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #126522 ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^11|c^2=e,d^2=g,e^2=h,f^2=ik,g^2=j,h^2=j,i^2=k,b^a=bf,c^a=cf,c^b=cfi,d^a=dik,d^b=dik,d^c=dj,e^a=ei,e^b=eik,f^a=fi,f^b=fi,f^c=fk,g^a=gk,g^b=gk,h^a=hk,h^b=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258190
maximal normal subgroup of	#72258190
maximal quotient of	#72258148

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