#72234278

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₃₂ · Schur multiplier (H₂) = C₂×C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆²	C₂²
commutator	C₂	C₂×C₈×C₃₂
Frattini, cosocle	C₈×C₁₆	C₂³
socle	C₂²	C₂×C₈×C₁₆
prime radical	C₂³	C₈×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆² ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

640 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234281, #72234292, #72234295
maximal normal subgroup of	#72234281, #72234292, #72234295
maximal quotient of	#72234297

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