#72287168

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,c^2=f,d^2=fh,e^2=gj,f^2=h,g^2=i,i^2=j,c^a=chj,d^a=dh,d^c=dh,e^a=ej,e^b=ehj,e^c=eg,e^d=ei,f^e=fj,g^c=gi,g^d=gj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287431, #72287589, …
maximal normal subgroup of	#72287278, #72287431, #72287589, …

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