#72258115

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

8627608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 9 generators on 32 points:

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

Polycyclic presentation:

2^11|h^2=j,i^2=jk,f^a=fj,g^a=gjk,h^a=hk,h^b=hk,h^c=hjk,h^d=hk,h^e=hj,i^a=ij,i^b=ij,i^c=ik,i^d=ijk,i^e=ik

Conjugacy classes & character table

536 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271037, #72301510, #72301985
maximal normal subgroup of	#72271037, #72301510, #72301985

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