#72321754

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	#24248
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#576	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3264 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)(5,6,10,16,9,19,11,18,12,14,29,35,30,15,32,17)(7,20,22,8,24,27,23,26,13,28,33,25,21,31,34,36);(2,3)(5,7,9,21,12,13,30,24)(6,8,15,26,14,25,19,36)(10,23,11,22,29,34,32,33)(16,28,17,31,35,20,18,27);(1,2)(3,4)(5,8)(6,13,14,7)(9,26)(10,31)(11,20)(12,25)(15,24,19,21)(16,22,35,33)(17,34,18,23)(27,29)(28,32)(30,36);(1,3,4,2)(6,14)(7,13)(8,25)(15,19)(16,35)(17,18)(20,28)(21,24)(22,33)(23,34)(26,36)(27,31)

Polycyclic presentation:

2^10|a^2=f,b^2=e,c^2=eij,d^2=fhj,e^2=j,f^2=i,g^2=ij,h^2=ij,i^2=j,b^a=bfij,c^a=cfhi,c^b=cfhi,d^a=defgij,d^b=dghij,d^c=defhij,e^a=ej,f^b=fi,f^c=fj,g^a=gj,g^b=gij,h^b=hij,h^c=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321803, #72322831
maximal normal subgroup of	#72321803, #72322831

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