#72316735

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1898
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	C₂×#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=egi,c^2=j,e^2=j,f^2=i,b^a=be,c^a=ci,c^b=cij,d^b=df,d^c=dj,e^a=eh,e^b=ehj,f^b=fi,f^d=fi,g^b=ghj,g^e=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316767, #72316779, #72316803
maximal normal subgroup of	#72316767, #72316779, #72316803

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