#72314491

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#62251	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5420 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:

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

Polycyclic presentation:

2^10|c^2=hj,e^2=f,f^2=ij,g^2=ij,i^2=j,c^a=ch,c^b=chj,d^b=dhj,e^a=ehj,e^b=ej,e^c=eg,e^d=eg,f^c=fi,f^d=fi,g^c=gi,g^d=gi,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314497, #72314559, #72315974
maximal normal subgroup of	#72314497, #72314559, #72315974

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