#72239799

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#3109
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33688	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #20331 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3249 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=ghj,d^2=fhij,e^2=h,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bij,d^a=df,d^b=di,d^c=dgh,e^a=ei,e^b=ej,f^a=fi,f^b=fj,f^c=fj,g^d=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239846
maximal normal subgroup of	#72239846

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