#72339827

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#12697
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	#10575552	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#117 ◁ #62815 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4570 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)(3,12)(4,18)(5,23)(6,10)(7,11)(8,15)(9,19)(13,17)(14,27)(16,25)(20,22)(21,26)(24,31)(28,29)(30,32);(1,3,4,14)(2,8,9,24)(5,16,22,30)(6,15,21,31)(7,17,19,29)(10,13,20,28)(11,25,18,32)(12,26,27,23);(4,19)(9,18)(14,29)(20,26)(21,22)(24,32)(27,28)(30,31);(1,2)(3,13)(4,20)(5,10)(6,23)(7,11)(8,15)(9,21)(12,17)(14,24)(16,25)(18,22)(19,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^10|b^2=e,f^2=j,g^2=i,h^2=j,b^a=bf,c^b=cij,d^b=dg,e^a=eij,e^c=ej,e^d=eh,f^a=fj,f^b=fi,f^e=fj,g^b=ghij,g^d=gi,g^e=gi,h^b=hi,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339829, #72339838, #72339845
maximal normal subgroup of	#72339829, #72339838, #72339845

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