#72319633

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₂²×C₁₆	C₂⁴
prime radical	#512914	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ #12662 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(6,12)(7,18)(8,19)(9,22)(10,25)(13,24)(14,23)(15,31)(16,34)(17,35)(20,32)(21,30)(26,36)(27,29)(28,33);(1,2,3,4);(2,4)(5,6,13,28,9,16,23,30,25,21,20,34,36,33,8,12,11,7,19,29,26,15,32,35,10,17,14,31,22,27,24,18);(5,7,20,34,26,16,24,18,25,17,19,29,22,33,23,30,11,6,14,31,9,15,8,12,10,21,13,28,36,27,32,35)

Polycyclic presentation:

2^10|b^2=e,c^2=fhj,d^2=fgi,f^2=hi,g^2=ij,h^2=ij,i^2=j,c^a=cf,c^b=ce,d^a=df,d^c=di,f^a=fh,f^d=fj,g^a=gi,g^c=gj,g^d=gj,h^a=hi,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#13658
maximal normal subgroup of	D₄×#13658

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