#72336444

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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 9632 conjugacy classes of subgroups; only the 3085 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)(3,8)(4,5)(6,7)(17,18)(19,23)(20,21)(22,24)(26,27)(28,32);(1,3,6,4)(2,5,7,8)(9,10,13,11)(12,14,16,15)(26,27)(30,31);(9,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24);(3,4)(5,8)(10,11)(14,15)(18,20)(22,23)(25,26,29,27)(28,30,32,31);(1,4,6,3)(2,5,7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,27,29,26)(28,31,32,30);(1,5)(2,4)(3,7)(6,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^10|b^2=h,d^2=i,e^2=hi,f^2=ghij,b^a=bg,c^b=cgh,d^a=dj,d^b=dhi,d^c=dghij,e^a=ej,e^b=ei,e^c=eghij,e^d=eh,f^a=fj,f^c=fghij,f^d=fgij,f^e=fh

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336494, #72336654
maximal normal subgroup of	#72336494, #72336654

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