#72329162

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 9671 conjugacy classes of subgroups; only the 3037 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)(3,6)(4,7)(5,8)(9,10,14,12)(11,13,15,16)(17,18,21,20)(19,23,24,22)(26,27)(30,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(25,26)(27,29)(28,30)(31,32);(10,12)(11,15)(17,19)(18,22)(20,23)(21,24)(25,26,29,27)(28,30,32,31);(10,12)(13,16)(18,20)(22,23)(26,27)(30,31);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,16,15,13)(17,19)(18,22)(20,23)(21,24)(25,27,29,26)(28,30,32,31);(1,2)(3,5)(4,7)(6,8)(9,13,14,16)(10,15,12,11)(17,20,21,18)(19,23,24,22)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^10|a^2=h,c^2=hij,e^2=g,f^2=h,b^a=bhi,c^a=cij,c^b=cg,d^a=dh,d^b=dhij,d^c=dhij,e^a=egh,e^b=ehij,e^c=eghij,e^d=eg,f^d=fh,f^e=fhi

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329242
maximal normal subgroup of	#72329242
maximal quotient of	#72331256

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