#72336845

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 9817 conjugacy classes of subgroups; only the 3021 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=hj,c^2=gi,e^2=hj,f^2=ghij,b^a=bg,c^a=cghi,c^b=cghj,d^a=dh,d^b=dhj,d^c=dgi,e^a=ej,e^c=eghij,e^d=ehj,f^a=fj,f^c=fghij,f^d=fgj,f^e=fhi

Conjugacy classes & character table

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