#72233840

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#23984
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂⁵
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 8992 conjugacy classes of subgroups; only the 664 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=h,b^2=hi,f^2=k,g^2=j,c^a=cg,c^b=cf,d^a=dh,d^b=dk,e^b=ei,e^c=ek,f^a=fi,f^b=fj,f^c=fk,f^e=fk,g^b=gik,g^c=gj,g^d=gj,g^f=gk,h^c=hj,i^c=ik

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234105
maximal normal subgroup of	#72234105

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