#72278177

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 21050 conjugacy classes of subgroups; only the 5992 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2)(3,4)(5,6)(7,8)(18,19);(1,2,3,4)(5,6,7,8)(17,18,20,19);(9,10,11,12)(14,16)(18,19);(9,10)(11,12)(13,14)(15,16)(17,19,20,18);(5,6)(7,8)(9,10,11,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)

Polycyclic presentation:

2^11|b^2=i,c^2=g,d^2=k,e^2=g,b^a=bi,c^b=ck,d^a=dk,d^c=dh,e^b=ej,e^d=eg,f^b=fik,f^c=fg,f^e=fg

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278177, #72298631, #72336150
maximal normal subgroup of	C₂×#72278177, #72298631, #72336150
maximal quotient of	C₂×#72278177, #72299933

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);