#72336330

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 9512 conjugacy classes of subgroups; only the 3013 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)(3,5)(4,6)(7,8)(10,13)(11,15)(18,19)(20,24)(26,27)(28,32);(1,2,4,6)(3,5,7,8)(17,18)(19,21)(20,22)(23,24)(25,26,29,27)(28,30,32,31);(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,31)(30,32);(11,15)(14,16)(18,19)(22,23)(26,27)(28,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,22,24,23)(25,27,29,26)(28,30,32,31);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^10|b^2=gh,e^2=gi,f^2=hi,b^a=bg,c^a=cj,c^b=cghi,d^b=dgi,d^c=dgi,e^a=ej,e^b=ehi,e^c=egi,e^d=egi,f^a=fj,f^c=fgh,f^d=fhj,f^e=fij

Conjugacy classes & character table

112 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, 2, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);