#72250071

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#30396
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#498812	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1897 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)(6,12,16,17)(8,22)(13,26,23,28)(14,24)(15,31,19,27)(20,34)(29,36,32,35)(30,33);(2,4)(6,13,16,23)(7,18)(8,14)(9,10)(12,26,17,28)(15,32,19,29)(20,33)(22,24)(27,36,31,35)(30,34);(5,6,14,28,9,12,8,23,11,16,24,26,10,17,22,13)(7,19,30,35,25,31,20,29,18,15,33,36,21,27,34,32);(1,2,3,4)(5,7,10,21,11,18,9,25)(6,15,17,31,16,19,12,27)(8,20,14,30,22,34,24,33)(13,29,26,35,23,32,28,36)

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=ehi,d^2=fhi,e^2=i,g^2=j,i^2=j,c^a=cg,c^b=ce,d^a=df,d^b=dfghi,d^c=dj,e^a=ej,e^b=ei,e^c=ej,f^c=fj,g^b=gj,g^d=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250127, #72257186, #72257392
maximal normal subgroup of	#72250127, #72257186, #72257392

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