#72277893

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₁₆² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₁₆	C₄²
commutator	C₄	C₁₆²
Frattini, cosocle	C₈×C₃₂	C₂²
socle, prime radical	C₂²	#7329
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 2 generators on 80 points:

(17,18,19,25,20,27,31,46,21,28,32,47,34,50,55,71,22,29,33,44,35,51,56,67,37,52,57,68,58,73,75,79,23,26,30,45,36,48,53,69,38,49,54,70,59,72,74,80,39,24,40,41,60,42,62,64,61,43,63,65,76,66,77,78);(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,24,26,29)(19,30)(25,44,45,41)(27,42,48,51)(28,43,49,52)(31,53)(32,54)(33,40)(46,67,69,64)(47,68,70,65)(50,66,72,73)(55,74)(56,62)(57,63)(71,79,80,78)(75,77)

Polycyclic presentation:

2^10|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^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

Frattini subgroup of	#72277967
maximal subgroup of	#72277905, #72277936, #72277944
maximal normal subgroup of	#72277905, #72277936, #72277944
maximal quotient of	#72277936

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.8);;
gap> SetPower(coll, 7, 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);