#72343279

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7622812	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,19,14,18,28,21)(11,22,23,16,24,12,25,26)(15,31,29,20,30,38,34,37)(17,33,32,27,35,40,36,39);(9,11)(10,16)(12,19)(13,23)(14,24)(15,32)(17,34)(18,26)(20,39)(21,22)(25,28)(27,37)(29,35)(30,36)(31,40)(33,38);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,33)(18,35)(19,36)(21,32)(23,31)(24,37)(25,38)(26,28)(27,34)(29,39)(30,40);(2,6)(5,8)(10,18)(12,22)(16,26)(19,21)(20,37)(27,39)(31,38)(33,40);(2,6)(3,7)

Polycyclic presentation:

2^10|a^2=f,f^2=j,h^2=i,i^2=j,b^a=bij,c^a=ch,c^b=chj,d^a=dg,d^c=dj,e^a=egj,e^c=egj,f^c=fj,h^a=hi,h^b=hij,h^c=hij,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343283
maximal normal subgroup of	#72343283
maximal quotient of	#72343866

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