#72328702

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 8778 conjugacy classes of subgroups; only the 3133 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:

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

Polycyclic presentation:

2^10|e^2=i,f^2=h,b^a=bj,c^b=cg,e^a=egj,e^b=eh,e^d=egj,f^a=fgi,f^b=fh,f^d=fgh

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328746
maximal normal subgroup of	#72328746
maximal quotient of	#72331614

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