#72339826

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#8522
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₄×M₁₆	C₂⁴
prime radical	#10575552	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4342 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)(4,9)(5,21)(6,10)(7,11)(8,16)(13,14)(15,29)(17,23)(18,25)(19,22)(20,26)(24,31)(27,28)(30,32);(1,3,4,15,6,17,20,31,7,13,18,28,5,16,22,32)(2,8,9,24,10,12,26,27,11,23,25,30,21,14,19,29);(3,13)(4,18)(8,23)(9,25)(12,14)(16,17)(19,26)(20,22);(1,2)(3,14)(4,19)(5,10)(6,21)(7,11)(8,16)(9,22)(12,13)(15,30)(17,23)(18,26)(20,25)(24,28)(27,31)(29,32)

Polycyclic presentation:

2^10|b^2=e,e^2=hj,f^2=j,g^2=i,h^2=j,b^a=bf,c^b=ci,d^b=dg,e^a=ei,e^c=ej,e^d=eh,f^a=fj,f^b=fij,f^e=fj,g^b=ghij,g^d=gi,h^d=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339829, #72339834, #72339850, …
maximal normal subgroup of	#72339829, #72339834, #72339850, …

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