#72336442

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
prime radical	#647656	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8834 conjugacy classes of subgroups; only the 3161 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:

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

Polycyclic presentation:

2^10|d^2=ij,e^2=h,f^2=gh,b^a=bg,c^b=ci,d^a=dj,d^c=dgh,e^a=ej,e^c=egh,e^d=ehij,f^a=fj,f^c=fgh,f^d=fgij,f^e=fhij

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336494, #72336654
maximal normal subgroup of	#72336494, #72336654

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