#72232680

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 9815 conjugacy classes of subgroups; only the 3013 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,14,16,15)(17,18)(19,24)(20,22)(21,23)(25,26)(27,29)(28,30)(31,32);(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,24,21,23)(25,26,30,28)(27,31,32,29);(4,8)(6,7)(12,16)(14,15)(17,19,22,21)(18,23,20,24)(25,27,30,32)(26,31,28,29);(2,3)(6,7)(10,11)(14,15)(18,20)(23,24)(26,28)(29,31);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,14,16,15)(17,20,22,18)(19,24,21,23)(25,28,30,26)(27,29,32,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24)(25,29)(26,32)(27,28)(30,31)

Polycyclic presentation:

2^10|a^2=h,b^2=hj,c^2=g,e^2=j,f^2=hi,b^a=bj,c^a=cg,c^b=ci,d^a=dj,d^b=dj,e^a=ehj,e^b=eh,e^d=ej,f^a=fgi,f^b=fhij,f^c=fij,f^d=fgi,f^e=fgh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710
maximal normal subgroup of	#72232710

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