#72262013

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

8588366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^11|f^2=jk,g^2=j,d^b=dj,e^b=ejk,f^a=fjk,g^a=gj,h^b=hk,h^c=hjk,h^f=hjk,h^g=hk,i^b=ijk,i^c=ij,i^f=ik,i^g=ij

Conjugacy classes & character table

536 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72262224
maximal subgroup of	#72262224
maximal normal subgroup of	#72262224
automorphism group of	C₂×#344

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