#72265614

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2557
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	#15463
prime radical	C₂³×#41	C₂³
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=g,c^2=eg,d^2=h,e^2=j,b^a=be,c^a=cg,c^b=cei,d^a=dg,d^b=dh,d^c=df,e^a=ej,e^b=ej,e^c=ej,f^a=fj,g^c=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265627, #72314015, #72314310
maximal normal subgroup of	#72265627, #72314015, #72314310

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