#72340835

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#147559
commutator	C₂²×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#532	C₂³
prime radical	#6443311	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3369 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:

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

Polycyclic presentation:

2^10|b^2=f,f^2=j,g^2=ij,h^2=j,b^a=bd,c^a=cde,c^b=cg,d^b=di,d^c=dj,e^c=ej,f^a=fi,f^c=fh,f^d=fj,g^a=gij,g^b=ghi,g^c=gij,g^f=gi,h^a=hj,h^b=hi,h^c=hj,h^f=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340840, #72340843, #72340859
maximal normal subgroup of	#72340840, #72340843, #72340859

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