#72265392

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2670
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	#16255
prime radical	C₂²×#575	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5733 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)(5,6,10,14)(7,15,18,31)(8,19,23,12)(9,24,21,13)(11,25,28,17)(16,29,34,33)(20,27,35,36)(22,30,26,32);(5,7)(6,11)(8,20,23,35)(9,25)(10,18)(12,29,19,33)(13,31)(14,28)(15,24)(16,26,34,22)(17,21)(27,32,36,30);(5,8,21,22,10,23,9,26)(6,12,24,30,14,19,13,32)(7,16,17,35,18,34,25,20)(11,27,15,33,28,36,31,29);(1,2,3,4)(11,28)(12,19)(15,31)(30,32)

Polycyclic presentation:

2^10|a^2=j,b^2=i,c^2=egi,d^2=hij,e^2=j,b^a=be,c^a=ci,c^b=ceg,d^a=dgh,d^b=df,d^c=df,e^a=ej,e^b=ej,e^c=ej,f^a=fj,g^b=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265401, #72314069, #72314310
maximal normal subgroup of	#72265401, #72314069, #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.9);;
gap> SetPower(coll, 3, F.5*F.7*F.9);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
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, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);