#72278321

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

22019 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=g,d^2=j,e^2=ghj,b^a=bh,c^b=cj,d^a=dj,d^c=dhj,e^a=ehj,e^b=ei,e^d=eg,f^b=fhj,f^c=fg,f^e=fghj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72278574, #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, 2, F.8);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);