#72224795

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁶
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

33860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^11|b^2=h,c^2=hj,d^2=h,f^2=i,g^2=j,i^2=k,c^a=chj,c^b=cg,d^c=dj,e^b=ek,f^a=fk,f^b=fi,f^c=fi,f^d=fk,g^b=gj,g^c=gj,i^b=ik,i^c=ik

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226832
maximal normal subgroup 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, 2, F.8);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);