#72314286

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	#7622820	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ #20384 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4185 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)(3,8)(4,7)(5,6)(9,10)(11,13)(12,19)(14,17)(15,18)(16,24)(20,27)(21,28)(22,32)(23,35)(25,37)(26,33)(29,38)(30,39)(31,36)(34,40);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,22)(14,29,27,36)(15,18,28,21)(16,32)(17,31,20,38)(19,30)(23,34)(24,39)(25,35)(26,37)(33,40);(2,6)(3,7)(9,12)(10,16)(11,19)(13,24)(14,30)(15,23)(17,22)(18,33)(20,39)(21,35)(25,38)(26,28)(27,32)(29,34)(31,40)(36,37);(1,4)(2,7)(3,6)(5,8)(10,11)(16,19)(17,20)(18,21)(22,39)(29,36)(33,35)(34,37);(2,6)(4,8)(12,23,24,26)(14,27)(16,33,19,35)(17,20)(22,34,39,37)(25,32,40,30)(29,36)(31,38)

Polycyclic presentation:

2^10|b^2=f,e^2=fj,h^2=i,i^2=j,b^a=bg,c^a=cg,c^b=cghij,d^a=dj,d^b=dg,e^a=egj,e^c=ei,e^d=egj,f^c=fj,h^b=hi,h^c=hij,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314371, #72314521, #72315218
maximal normal subgroup of	#72314371, #72314521, #72315218

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