#72277918

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₄²
commutator	C₄	C₈×C₃₂
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

85 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277936, #72277944
maximal normal subgroup of	#72277936, #72277944

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, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
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);