#72262844

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=ei,c^2=efg,e^2=j,b^a=be,c^a=cehj,c^b=cf,d^a=dgij,d^b=dg,e^a=ej,e^c=eh,e^d=ei,f^a=fhj,f^b=fh,f^c=fhi,f^d=fi,f^e=fj,g^b=gi,g^c=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262846, #72262858, #72262878, …
maximal normal subgroup of	#72262846, #72262858, #72262878, …
maximal quotient of	#72339062, #72339065

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