#72250040

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	#24338
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,c^2=f,d^2=gi,e^2=i,g^2=ij,i^2=j,b^a=bh,c^a=cfh,c^b=ch,d^a=de,d^b=dg,d^c=deg,e^a=ei,e^b=eij,e^c=ej,f^d=fj,g^a=gij,g^b=gi,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250103, #72256845, #72256936
maximal normal subgroup of	#72250103, #72256845, #72256936

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