#72234280

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄²×C₃₂ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₆	C₂²
commutator	C₂	C₄²×C₃₂
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
socle, prime radical	C₂³	C₂×C₄×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(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);(5,7,8,13,9,14,15,19)(6,10,12,17,11,16,18,20);(1,2,3,4)(6,11)(10,16)(12,18)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

640 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234289, #72234297, #72234303
maximal normal subgroup of	#72234289, #72234297, #72234303
maximal quotient of	#72234281, #72234306, #72234371

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