#72328667

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 9638 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:

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

Polycyclic presentation:

2^10|b^2=hj,c^2=i,f^2=hij,b^a=bh,c^a=ci,c^b=cg,d^b=dhj,d^c=di,e^a=egh,e^b=egh,e^c=egh,e^d=egh,f^a=fi,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	#72328746
maximal normal subgroup of	#72328746
maximal quotient of	#72331497

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