#72321796

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31996
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#532	D₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=f,d^2=ij,e^2=j,g^2=i,h^2=i,i^2=j,b^a=bhij,c^a=cg,c^b=cf,d^b=dghij,d^c=degh,e^a=ej,f^d=fj,g^b=gi,g^c=gij,g^d=gj,h^b=hi,h^c=hij,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321841, #72322591, #72322802
maximal normal subgroup of	#72321841, #72322591, #72322802

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