#72234310

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₁₆×C₃₂ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈×C₁₆	C₂²
commutator	C₂	C₁₆×C₃₂
socle, prime radical	C₂³	C₈×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 2 generators on 52 points:

(1,2)(3,4)(21,22,23,27,24,28,31,37,25,29,32,38,34,40,43,47,26,30,33,39,35,41,44,48,36,42,45,49,46,50,51,52);(2,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20)

Polycyclic presentation:

2^10|a^2=c,b^2=d,c^2=f,d^2=g,f^2=h,g^2=i,h^2=j,b^a=be

Conjugacy classes & character table

640 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234325
maximal normal subgroup of	#72234325
maximal quotient of	#72234312, #72234410, #72234411

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