#72320573

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#12674
commutator	C₂×C₁₆	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
socle	C₂	#129154
prime radical	C₂³	#2179
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ #11399 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=d,c^2=fhij,d^2=g,e^2=gij,f^2=hij,g^2=j,h^2=ij,i^2=j,b^a=bghi,c^a=cij,c^b=cf,d^a=dj,d^c=dj,e^b=ei,e^c=ej,f^a=fj,f^b=fh,h^b=hi,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320581, #72320660, #72320998, …
maximal normal subgroup of	#72320581, #72320660, #72320998, …

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