#72314577

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	#7623099	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20437 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2953 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=fhj,d^2=h,e^2=gh,g^2=i,i^2=j,b^a=bj,c^a=chj,c^b=chj,d^a=dh,d^b=dj,d^c=dj,e^a=ehj,e^b=eg,e^c=ei,e^d=egi,f^e=fj,g^b=gi,g^c=gj,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314594, #72314619, #72315390
maximal normal subgroup of	#72314594, #72314619, #72315390

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