#72336429

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9562 conjugacy classes of subgroups; only the 3061 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=hj,c^2=gi,d^2=ij,f^2=ghij,b^a=bg,c^a=cghi,c^b=cghj,d^a=dhij,d^b=dhi,d^c=dgj,e^a=eij,e^b=eij,e^c=eij,e^d=eij,f^a=fj,f^c=fghij,f^d=fgi,f^e=fhj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336494
maximal normal subgroup of	#72336494

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