#72304133

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 7401 conjugacy classes of subgroups; only the 2973 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=g,d^2=ij,b^a=bgi,c^a=cghj,c^b=ch,d^a=dgi,d^b=dghi,d^c=dgh,e^b=ej,e^c=eghj,e^d=ej,f^a=fi,f^b=fi,f^c=fgj,f^d=fi

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