#72343984

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×#3416	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2496 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=hj,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bi,c^a=chj,d^a=dg,d^b=dgh,d^c=dj,e^c=ej,f^d=fj,g^a=gi,g^b=gij,g^d=gij,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343989
maximal normal subgroup of	#72343989

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