#72307017

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2023
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=gj,c^2=fhi,e^2=i,f^2=i,i^2=j,b^a=bg,c^a=ce,c^b=cefij,d^c=dgh,e^a=ei,e^b=ej,e^c=ej,f^a=fi,f^c=fj,g^c=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307029, #72307077, #72307266
maximal normal subgroup of	#72307029, #72307077, #72307266

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