#72327893

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

24145 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,e^2=h,b^a=bg,c^a=cj,c^b=cghi,d^a=dh,d^b=dh,e^a=ehj,e^c=ej,e^d=eh,f^a=fj,f^b=fhij,f^d=fhij,f^e=fghij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327940, #72328255
maximal normal subgroup of	#72327940, #72328255

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