#72224901

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#11049
commutator	C₂×C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂³
prime radical	C₂×#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efgi,c^2=efghi,d^2=h,e^2=i,f^2=ij,g^2=j,i^2=j,b^a=bf,c^a=cfgh,c^b=cd,d^b=dh,d^c=dh,e^b=ei,e^c=ei,f^b=fij,f^c=fi,g^a=gj,g^b=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224904, #72224940, #72226820
maximal normal subgroup of	#72224904, #72224940, #72226820

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