#72285486

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1790
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#7708257	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=j,c^2=e,d^2=g,e^2=j,c^a=cf,c^b=ci,d^a=df,d^b=di,d^c=deg,e^a=eh,f^c=fh,f^d=fi,f^e=fj,g^a=gi,g^c=gj,h^c=hj,h^d=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285624, #72306394, #72306409
maximal normal subgroup of	#72285624, #72306394, #72306409

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