#72240216

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14715
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=f,d^2=h,e^2=i,h^2=j,b^a=bij,c^a=cf,c^b=cg,d^a=dh,d^c=de,e^b=ej,f^d=fi,g^d=gj,h^a=hj,h^c=hi

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240248
maximal normal subgroup of	#72240248
maximal quotient of	#72240259, #72240274

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