#72231738

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#2403
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁵
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

At least 8571 conjugacy classes of subgroups; only the 1788 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^11|b^2=i,e^2=fgk,f^2=k,g^2=k,h^2=k,b^a=bi,d^b=di,e^a=ef,e^b=eh,e^c=eg,e^d=efgk,f^a=fk,f^b=fj,f^d=fk,g^c=gk,g^d=gk,h^a=hk,h^b=hj,h^d=hk,h^e=hjk,i^e=ijk

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

automorphism group of

#1276

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