#72343449

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#532	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ C₂×#2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3648 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fi,f^2=j,h^2=i,i^2=j,b^a=bh,d^c=dg,e^a=ej,e^c=ej,e^d=ej,f^b=fj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343466, #72343493, #72343764
maximal normal subgroup of	#72343466, #72343493, #72343764

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