#72278448

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

28829 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=hij,b^a=bgh,c^b=chi,d^b=dghij,d^c=dg,e^a=ehi,e^c=eghi,e^d=eghj,f^a=fghij,f^c=fhj,f^d=fhij,f^e=fg

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72278489
maximal subgroup of	#72278489
maximal normal subgroup of	#72278489
maximal quotient of	#72299691

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