#72313855

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33717	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20476 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=f,d^2=f,e^2=h,h^2=i,i^2=j,c^a=cj,c^b=cgj,d^a=dgj,d^b=dj,e^c=eh,e^d=ei,f^e=fj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313898, #72314059, #72314202
maximal normal subgroup of	#72313898, #72314059, #72314202

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