#72334174

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 8930 conjugacy classes of subgroups; only the 3045 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:

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

Polycyclic presentation:

2^10|b^2=h,e^2=ghij,b^a=bi,c^b=cg,d^b=dgij,e^a=eij,e^b=egij,e^c=eij,e^d=egij,f^b=fj,f^c=fij,f^d=fhj,f^e=fh

Conjugacy classes & character table

124 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.8);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);