#72225347

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#7292
commutator	C₂×C₁₆	C₄×C₈
Frattini, cosocle	C₄×#346	C₂²
socle	C₂	#93458
prime radical	C₂³	#1385
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=c,c^2=f,d^2=gh,e^2=hij,f^2=ij,g^2=i,i^2=j,b^a=bd,c^a=cg,d^a=dg,d^b=dh,d^c=dj,e^b=ehij,f^a=fi,g^a=gij,g^b=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225348, #72225368, #72225370, …
maximal normal subgroup of	#72225348, #72225368, #72225370, …

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