#72338744

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#480250
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	#379	C₂⁴
prime radical	C₂×#3484	C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #34171 ◁ G
LECS	G ▷ #379 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

3075 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:

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

Polycyclic presentation:

2^10|a^2=e,d^2=ghj,e^2=g,b^a=bghij,c^a=cgij,d^a=df,d^c=dj,e^b=eij,e^c=ej,e^d=eh,f^a=fh,f^c=fj,f^d=fi,f^e=fi,g^d=gj,h^a=hi,h^d=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338745, #72338750, #72338796, …
maximal normal subgroup of	#72338745, #72338750, #72338796, …

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