#72278166

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

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

38149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=g,e^2=fg,b^a=bi,d^c=dh,e^a=ef,e^b=ej,e^d=eg

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278416, #72335761, #72335803
maximal normal subgroup of	#72278416, #72335761, #72335803
maximal quotient of	#72299924

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