#72250573

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1783 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=efj,d^2=efgi,e^2=i,f^2=j,h^2=j,i^2=j,c^a=cghj,c^b=ce,d^a=dh,d^b=defgi,d^c=dij,e^a=ej,e^b=ei,e^c=ej,f^b=fj,f^c=fj,f^d=fj,g^c=gj,g^d=gj,h^b=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250632, #72250983, #72252664
maximal normal subgroup of	#72250632, #72250983, #72252664

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