#72322310

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂×#2177
prime radical	C₂×#2263	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(10,18)(11,25)(12,30)(13,29)(14,15)(16,27)(19,37)(20,32)(21,26)(22,34)(23,35)(24,40)(28,33)(31,38)(36,39);(1,2,4,3,5,6,8,7)(9,10,12,21,14,22,13,32,27,39,33,37,25,19,38,36,17,20,31,34,11,26,28,18,16,24,29,35,15,23,30,40);(1,3,4,6,5,7,8,2)(9,11,27,15,17,14,16,25)(10,19,24,22,20,23,39,26)(12,13,33,38,31,28,29,30)(18,34,36,37,32,21,40,35);(10,20)(18,32)(19,23)(21,34)(22,26)(24,39)(35,37)(36,40)

Polycyclic presentation:

2^10|b^2=e,c^2=ef,e^2=g,f^2=hj,g^2=ij,h^2=i,i^2=j,b^a=bf,c^a=ci,c^b=cghij,d^b=dj,e^a=eh,e^c=ej,f^a=fhi,f^b=fj,f^c=fj,g^a=gi,h^a=hij,i^a=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322316, #72322334, #72322349, …
maximal normal subgroup of	#72322316, #72322334, #72322349, …

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