#72278050

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₄²
commutator	C₄	C₄×C₆₄
Frattini, cosocle	C₄×C₆₄	C₂²
socle, prime radical	C₂²	#7510
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₄×C₆₄ ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

79 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 2 generators on 136 points:

(2,5)(6,8)(9,10,12,21,14,23,38,63,16,25,40,65,49,73,96,117,17,26,41,66,50,74,97,118,34,60,83,61,86,110,127,112,18,27,42,67,51,75,98,119,53,77,89,113,103,123,129,135,47,22,69,62,102,70,121,114,35,58,36,88,87,108,90,128,19,28,43,68,52,76,99,120,54,78,91,111,104,124,130,134,46,71,94,115,101,122,132,136,33,20,55,56,85,57,105,106,15,24,39,64,48,72,95,116,31,59,81,107,84,109,126,133,13,44,37,92,45,100,93,131,11,29,30,79,32,80,82,125);(1,2,3,6,4,5,7,8)(9,11,13,31,15,33,46,54,19,35,47,53,18,34,17,16)(10,20,22,25,24,58,26,59,28,60,44,78,27,29,71,77)(12,36,37,89,39,83,94,40,43,30,69,81,42,55,41,91)(14,32,45,84,48,85,101,104,52,87,102,103,51,86,50,49)(21,61,62,111,64,79,66,113,68,56,92,65,67,88,115,107)(23,57,70,73,72,108,74,109,76,110,100,124,75,80,122,123)(38,90,93,129,95,127,132,96,99,82,121,126,98,105,97,130)(63,112,114,134,116,125,118,135,120,106,131,117,119,128,136,133)

Polycyclic presentation:

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

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

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