#72339870

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#246927
commutator, Frattini, cosocle	C₄×M₁₆	C₂⁴
LCS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1864 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)(5,10)(6,11)(7,15)(8,19)(9,23)(13,29)(14,28)(16,26)(17,25)(20,24)(21,22)(27,31)(30,32);(3,13,16,30,17,31,15,20)(4,19)(7,24,12,29,26,32,25,27)(8,18)(14,22)(21,28);(1,3,14,31,23,15,19,30,6,17,22,13,5,16,4,20)(2,7,21,32,9,25,18,29,11,26,28,24,10,12,8,27);(1,2)(3,7)(4,21)(5,9)(6,11)(8,14)(10,23)(12,15)(13,29)(16,25)(17,26)(18,22)(19,28)(20,32)(24,30)(27,31)

Polycyclic presentation:

2^10|b^2=g,c^2=eghj,e^2=h,f^2=j,g^2=ij,h^2=j,c^a=cf,c^b=ce,d^b=dgij,d^c=dghi,e^a=ei,e^b=ej,e^c=eh,e^d=ehj,f^a=fj,f^b=fij,f^c=fij,f^e=fj,g^c=gh,g^d=gij,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339877, #72339883
maximal normal subgroup of	#72339877, #72339883

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