#72319210

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×C₄×D₄
prime radical	C₂³×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₈×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,c^2=f,e^2=hj,f^2=i,h^2=j,b^a=bi,c^b=ch,d^a=di,e^b=eh,e^c=ej,f^b=fj,h^b=hj

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319221, #72319356, #72319571
maximal normal subgroup of	#72319221, #72319356, #72319571

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