#72314092

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	#7623719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20495 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2920 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=fj,h^2=i,i^2=j,b^a=bj,c^a=cj,c^b=ch,d^a=dgj,d^b=dgj,d^c=dg,e^c=egij,e^d=eg,f^c=fj,h^b=hi,h^c=hij,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314408, #72315218
maximal normal subgroup of	#72314098, #72314408, #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, 1, F.7);;
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.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
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);