#72279752

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2431
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	#14885
prime radical	#7670156	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

4862 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=g,d^2=f,e^2=i,f^2=j,h^2=ij,c^b=cg,d^a=dh,d^b=df,d^c=de,f^a=fij,f^b=fj,f^c=fi,g^d=gi,h^a=hj,h^b=hij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279961, #72280226, #72280374
maximal normal subgroup of	#72279961, #72280226, #72280374

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