#72334230

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 9570 conjugacy classes of subgroups; only the 3009 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:

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

Polycyclic presentation:

2^10|b^2=j,c^2=i,d^2=hj,e^2=h,b^a=bj,c^b=cg,d^b=dh,d^c=dhij,e^b=ehj,e^c=ehi,e^d=ej,f^a=fi,f^b=fh,f^c=fi,f^d=fgij,f^e=fghij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334244
maximal normal subgroup of	#72334244

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