#72313865

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33717	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20437 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fgj,d^2=g,e^2=h,h^2=i,i^2=j,b^a=bj,c^a=cgj,c^b=cgj,d^c=dg,e^b=eh,e^c=ei,f^e=fj,h^b=hi,h^c=hj,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313898, #72315346, D₄×#33471
maximal normal subgroup of	#72313898, #72315346, D₄×#33471

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