#72336818

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 10156 conjugacy classes of subgroups; only the 3061 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,23,24,22)(25,26,29,32)(27,30,31,28);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15);(3,8)(4,6)(9,11)(10,13)(12,16)(14,15)(18,20)(22,23)(25,26,29,32)(27,30,31,28);(10,14)(13,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,31)(30,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,27)(26,30)(28,32)(29,31);(1,4,5,6)(2,3,7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,31,32,27)

Polycyclic presentation:

2^10|a^2=h,c^2=gi,f^2=ghij,b^a=bg,c^a=cghi,c^b=cij,d^a=dh,d^c=dgi,f^a=fj,f^c=fghij,f^d=fg,f^e=fhij

Conjugacy classes & character table

118 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, 3, F.7*F.9);;
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.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
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);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);