#72263207

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 8784 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) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=i,d^2=hi,e^2=ij,c^a=chi,d^a=dg,d^b=dg,d^c=dj,e^a=ehi,e^b=eg,e^d=egi,f^a=fh,f^b=fh,f^c=fi,f^d=fgj,f^e=fij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263231
maximal normal subgroup of	#72263231

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