#72230722

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	C₈²
commutator	C₈	C₈×C₁₆
Frattini, cosocle	#7510	C₂²
socle	C₂	#94505
prime radical	C₂²	#7511
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ #7510 ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

74 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:

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

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=bh,c^b=cij,d^a=di,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	#72230725, #72230779, #72230781, …
maximal normal subgroup of	#72230725, #72230779, #72230781, …
maximal quotient 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.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
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);