#72321902

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	#32002
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#28180	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2038 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)(3,4)(6,12,13,22)(7,17)(8,21)(9,24)(10,18)(14,26,27,36)(15,16,28,25)(19,33)(23,34)(29,35,30,31);(2,4)(6,13)(12,22)(15,28)(16,25);(2,4)(5,6,10,25,11,13,18,16)(7,14,20,35,17,27,32,31)(8,22,21,15,9,12,24,28)(19,26,34,29,23,36,33,30);(5,7,18,32,11,17,10,20)(6,14,25,35,13,27,16,31)(8,23,24,34,9,19,21,33)(12,26,28,29,22,36,15,30)

Polycyclic presentation:

2^10|a^2=e,c^2=ei,d^2=ij,g^2=i,h^2=i,i^2=j,b^a=bf,c^a=cfgij,c^b=ceghij,d^a=dghi,d^b=de,d^c=deghij,e^c=ej,g^a=gi,g^c=gi,g^d=gj,h^a=hi,h^c=hij,i^a=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321926, #72322170
maximal normal subgroup of	#72321926, #72322170

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