#72322718

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	C₄×C₈	C₂⁵
prime radical	C₂×#3415	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17316 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,c^a=cj,d^a=dg,d^b=dh,e^c=ej,f^a=fj,g^b=gi,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322814, #72322990
maximal normal subgroup of	#72322814, #72322990

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