#72306804

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#13561
commutator	#127	C₂³×C₄
Frattini, cosocle	#493	C₂⁴
socle	C₂	C₂×#13561
prime radical	#6348073	C₂
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ #60639 ◁ G
LECS	G ▷ #493 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2061 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)(3,8)(4,15)(5,20)(6,11)(7,12)(9,19)(10,22)(13,27)(14,28)(16,25)(17,26)(18,23)(21,24)(29,31)(30,32);(1,3,7,6)(2,8,12,11)(4,16)(5,21,22,14)(9,25)(10,28,20,24)(13,29)(15,30)(17,18)(19,32)(23,31)(26,27);(1,4,5,16,6,18,14,29,7,19,22,32,3,13,21,17)(2,9,20,30,11,27,28,26,12,15,10,25,8,23,24,31);(5,22)(10,20)(14,21)(16,32)(17,29)(24,28)(25,30)(26,31)

Polycyclic presentation:

2^10|b^2=ij,c^2=ef,g^2=i,h^2=j,b^a=bi,c^a=cj,c^b=ce,d^b=di,d^c=dij,e^a=ej,e^b=ej,e^c=eg,e^d=ej,f^a=fj,f^b=fgij,f^c=fg,f^e=fhij,g^b=gi,g^c=gh,g^e=gi,g^f=gi,h^b=hi,h^c=hj,h^e=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306806, #72306832, #72306836, …
maximal normal subgroup of	#72306806, #72306832, #72306836, …

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