#72230728

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#322
commutator	C₈	C₄×C₃₂
Frattini, cosocle	#7510	C₂²
socle	C₂	#94508
prime radical	C₂²	#7511
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ #7510 ◁ G
LECS	G ▷ #7510 ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

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

Polycyclic presentation:

2^10|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=i,g^2=h,h^2=i,i^2=j,b^a=bfij,c^b=cij,d^a=dij,d^c=dj,e^b=ej,f^a=fj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230729
maximal normal subgroup of	#72230729

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