#72303277

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1790
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	C₂×#1790
prime radical	#7708257	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

5638 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:

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

Polycyclic presentation:

2^10|b^2=fg,c^2=j,d^2=h,e^2=j,b^a=bf,c^b=cj,d^a=dh,d^b=de,e^a=eh,e^b=eh,f^b=fi,f^d=fh,f^e=fj,g^a=gi,g^b=gi,g^d=gj,g^f=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303301, #72311144
maximal normal subgroup of	#72303301, #72311144

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