#72239770

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#7555
commutator	C₄³	C₂×C₄²
Frattini, cosocle	C₂²×C₄³	C₂³
prime radical	#7633352	C₂²
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄³ ▷ C₂³ ▷ C₁

5970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,10)(3,7,12,15)(4,9,5,8)(11,14,13,16)(17,18)(19,27)(20,24)(21,26)(22,25)(23,30)(28,31)(29,32)(35,42)(36,37)(39,43)(40,46)(44,45)(47,48);(1,3,4,11,6,12,5,13)(2,7,9,14,10,15,8,16)(17,19)(18,23,25,31)(20,28)(21,30)(22,29)(24,27,26,32)(33,34)(35,43)(36,40)(37,46)(38,41)(39,45)(42,47)(44,48);(2,8,10,9)(7,14,15,16)(18,24,25,26)(19,28,29,30)(33,35)(34,39)(36,45)(37,42)(38,44)(40,48)(41,47)(43,46)

Polycyclic presentation:

2^11|a^2=i,b^2=d,c^2=e,d^2=i,f^2=ik,g^2=j,h^2=k,b^a=bg,c^a=cf,c^b=ch,d^a=dj,e^a=eij,e^b=ej,f^a=fik,f^b=fij,f^c=fjk,g^a=gj,g^c=gjk,h^a=hik,h^b=hk,h^c=hjk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239794
maximal normal subgroup of	#72239794

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