#72309834

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21452
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33580	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ Q₈×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=h,d^2=ej,f^2=h,g^2=i,i^2=j,b^a=bg,c^a=chj,c^b=cf,d^a=dj,d^b=dij,d^c=dh,e^b=ej,f^b=fh,f^c=fh,g^a=gi,g^d=gj,i^a=ij

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309851, #72309925, #72309941, …
maximal normal subgroup of	#72309851, #72309925, #72309941, …

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