#72234014

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 9326 conjugacy classes of subgroups; only the 3005 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=hi,f^2=ij,c^b=cgh,d^a=dg,d^c=dhij,e^a=eh,e^b=ehij,e^c=ej,e^d=ej,f^c=fj,f^d=fij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234056
maximal normal subgroup of	#72234056

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