#72252841

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	C₂×C₄	C₂×#486
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	C₂×#2023
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gij,f^2=j,g^2=j,h^2=ij,i^2=j,b^a=bg,c^a=ce,c^b=cehi,d^b=dfgj,d^c=dfgj,e^b=ej,f^b=fj,g^b=gj,h^a=hj,h^b=hi,h^c=hi,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252852, #72252864, #72256849
maximal normal subgroup of	#72252852, #72252864, #72256849

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