#72280249

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#410
commutator	C₂²×C₄	C₂²×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂⁴
socle	C₂²	#20723
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

2357 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=g,c^2=e,d^2=hi,f^2=j,h^2=j,b^a=bij,c^a=ch,c^b=cf,d^b=dij,d^c=dhij,e^a=ej,e^b=ei,f^a=fj,f^b=fj,f^c=fij,f^d=fj,f^e=fj,i^c=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280297
maximal normal subgroup of	#72280297

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