#72336786

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

22263 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|b^2=gh,d^2=hj,f^2=gi,b^a=bg,c^b=cghi,d^a=dj,d^b=dgj,d^c=dgh,e^b=ehj,e^d=ehj,f^d=fghij,f^e=fhj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336887
maximal normal subgroup of	#72336887

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