#72260787

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

22211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=gh,d^2=j,e^2=j,b^a=bg,c^a=cgh,c^b=cg,d^a=dhi,d^b=di,d^c=dhi,e^a=ei,e^b=eh,e^c=eij,e^d=ej,f^b=fh,f^d=fj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260792, #72260850
maximal normal subgroup of	#72260792, #72260850

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