#72314028

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	D₄×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20495 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3096 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fhj,d^2=f,e^2=fghij,g^2=i,i^2=j,b^a=bh,c^a=ch,c^b=cj,d^b=dh,e^a=ehij,e^b=ej,e^c=ei,e^d=eg,f^e=fj,g^a=gj,g^c=gj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314059, #72314202, #72314329, …
maximal normal subgroup of	#72314059, #72314202, #72314329, …

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