#72319416

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#146661
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#127	C₂³
prime radical	C₂²×#453	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#117 ◁ #34210 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=d,c^2=efij,d^2=ij,e^2=ij,f^2=ij,h^2=j,b^a=bg,c^b=cdeij,d^a=dh,d^c=dj,e^a=eh,e^b=ej,f^a=fh,g^b=ghj,g^d=gij,g^e=gij,g^f=gij,h^a=hj,h^b=hi,h^g=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319428, #72319539, #72319635, …
maximal normal subgroup of	#72319428, #72319539, #72319635, …

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