#72314420

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	#7623723	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2677 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=f,d^2=hj,e^2=ghj,g^2=i,i^2=j,c^b=ch,d^a=dhj,d^b=dj,d^c=dj,e^a=ej,e^b=eg,e^c=eij,e^d=eghi,f^e=fj,g^b=gi,g^c=gj,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314441, #72314559, #72315244
maximal normal subgroup of	#72314441, #72314559, #72315244

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