#72321828

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31680
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#411643	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1720 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:

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

Polycyclic presentation:

2^10|b^2=e,d^2=gi,e^2=j,h^2=ij,i^2=j,b^a=bh,c^a=cf,c^b=cfg,d^a=dgh,d^b=dgi,d^c=defgij,e^a=ej,g^d=gj,h^a=hi,h^b=hij,h^d=hi,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321841, #72322386
maximal normal subgroup of	#72321841, #72322386

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