#72280949

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 8105 conjugacy classes of subgroups; only the 3325 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

(2,3)(5,6,9,11)(7,8,10,12)(14,17)(18,20)(22,26)(23,28);(1,2,4,3)(7,10)(8,12)(14,17)(15,19)(21,22,25,26)(23,27,28,24);(2,3)(13,14,16,17)(15,18,19,20)(21,22,25,26)(23,24,28,27);(1,3,4,2)(14,17)(18,20)(21,23,25,28)(22,27,26,24);(5,7,9,10)(6,8,11,12)(13,14)(15,20)(16,17)(18,19)(21,23)(22,27)(24,26)(25,28);(5,8)(6,10)(7,11)(9,12)(13,15,16,19)(14,18,17,20)(21,24,25,27)(22,28,26,23)

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bgh,c^a=cgi,c^b=chi,d^a=dgh,d^c=dhj,e^a=egi,e^b=ej,e^c=egi,e^d=egi,f^b=fi,f^e=fgi

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

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