#72304086

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	#846293	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6843 conjugacy classes of subgroups; only the 2993 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=g,d^2=j,e^2=j,c^a=cgh,c^b=ch,d^c=dgh,e^a=eg,e^b=eghi,e^c=eg,e^d=ej,f^a=fi,f^c=fhj,f^e=fij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304145
maximal normal subgroup of	#72304145

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