#72307239

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂×#2482
prime radical	#62814	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=gij,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,c^b=cj,d^c=dgh,e^a=eij,f^a=fi,f^c=fi,f^d=fj,h^a=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307250
maximal normal subgroup of	#72307250

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