#72231878

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#322
commutator	C₈	C₂²×C₄×C₁₆
Frattini, cosocle	#1311	C₂⁴
socle	C₂²	C₂²×#1313
prime radical	C₂×#41	#322
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #479899 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(19,29,30,51,32,53,33,54)(20,34,35,55)(21,37)(25,44,45,69,40,64,47,71)(26,48,41,66)(27,42)(31,57,58,78,59,77,56,52)(36,62,61,60)(38,63)(46,73,74,79,65,80,72,70)(49,76,67,75)(50,68);(17,18)(19,25)(20,26)(21,27)(22,28)(23,24)(29,44)(30,45)(31,46)(32,40)(33,47)(34,48)(35,41)(36,49)(37,42)(38,50)(39,43)(51,69)(52,70)(53,64)(54,71)(55,66)(56,72)(57,73)(58,74)(59,65)(60,75)(61,67)(62,76)(63,68)(77,80)(78,79);(18,24)(25,40)(26,41)(27,42)(28,43)(44,64)(45,47)(46,65)(48,66)(49,67)(50,68)(69,71)(70,79)(72,74)(73,80)(75,76);(17,19,20,31,21,29,36,52,22,33,34,58,38,54,60,57,23,32,35,59,37,53,61,78,39,30,55,56,63,51,62,77)(18,25,26,46,27,44,49,70,28,47,48,74,50,71,75,73,24,40,41,65,42,64,67,79,43,45,66,72,68,69,76,80)

Polycyclic presentation:

2^11|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^b=ck,d^a=dh,e^d=ej,f^a=fj,f^e=fk,g^d=gk,h^a=hk

Conjugacy classes & character table

356 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231918, #72231972, #72323120, …
maximal normal subgroup of	#72231918, #72231972, #72323120, …

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);