#72338397

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#1409
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	#493	C₂⁴
prime radical	#60647	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄ ◁ C₂×#41 ◁ C₄×#41 ◁ #60647 ◁ G
LECS	G ▷ #493 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|c^2=f,d^2=ghi,e^2=hi,g^2=ij,h^2=j,b^a=bj,c^a=cj,d^b=dj,d^c=de,e^c=eg,e^d=ej,f^d=fgh,f^e=fhi,g^c=gh,g^f=gij,h^c=hi,h^f=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338401, #72338404, #72338454
maximal normal subgroup of	#72338401, #72338404, #72338454

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