#72292977

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2127
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	C₂×#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #34070 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1853 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)(4,5)(6,8)(9,10,12,17)(11,20,21,37)(13,18,27,24)(14,15,25,30)(16,31,32,38)(19,26,35,28)(22,33,36,40)(23,29,39,34);(1,2)(3,7)(4,5)(6,8)(9,10,12,24)(11,20,36,32)(13,19,27,17)(14,15,25,35)(16,31,40,23)(18,26,30,28)(21,34,22,33)(29,39,37,38);(2,5)(6,7)(10,15)(12,25)(16,29)(18,19)(20,33)(21,36)(26,27)(31,39);(1,3,4,8)(2,6,5,7)(9,11)(10,16,15,29)(12,21)(13,23)(14,22)(17,34,30,32)(18,20,19,33)(24,40,35,37)(25,36)(26,31)(27,39)(28,38)

Polycyclic presentation:

2^10|a^2=e,b^2=efi,d^2=g,f^2=i,h^2=j,b^a=bf,c^a=cg,c^b=cg,d^a=dgh,d^b=dgh,e^b=eh,e^c=ej,e^d=ei,f^a=fhij,f^b=fh,f^d=fj,f^e=fi,g^a=gj,g^b=gj,h^a=hi,h^b=hij,h^e=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292978, #72292992, #72293082
maximal normal subgroup of	#72292978, #72292992, #72293082

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