#72250068

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#30396
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13323	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(6,12,15,16)(7,17,20,24)(8,21)(13,25,28,26)(14,29)(18,31)(19,32,34,30)(22,23);(5,6,8,13,9,16,22,25,11,15,21,28,10,12,23,26)(7,18,19,33,17,14,30,27,20,31,34,35,24,29,32,36);(2,4)(6,13,15,28)(7,19,20,34)(8,22)(9,10)(12,25,16,26)(14,31)(17,32,24,30)(18,29)(21,23)(27,36);(1,2,3,4)(5,7,8,19,9,17,22,30,11,20,21,34,10,24,23,32)(6,14,28,36,16,31,26,33,15,29,13,27,12,18,25,35)

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=h,d^2=efij,e^2=i,f^2=j,g^2=j,i^2=j,b^a=bg,c^b=ceg,d^a=df,d^b=dghj,d^c=defgh,e^a=ej,e^c=ei,f^b=fj,f^c=fj,g^c=gj,g^d=gj,h^b=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250127, #72257203, #72257420
maximal normal subgroup of	#72250127, #72257203, #72257420

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