#72239283

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	#36624	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄³ ▷ C₂³ ▷ C₁

5431 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)(3,10)(4,7)(5,8)(6,13)(9,14)(11,15)(12,16)(17,18)(19,26)(20,22)(21,24)(23,27)(25,30)(28,31)(29,32)(33,34)(35,42)(36,40)(37,41)(38,43)(39,46)(44,47)(45,48);(1,3,4,12,5,13,14,11)(2,6,7,15,8,10,9,16)(18,22,24,25)(19,27,28,29)(33,35)(34,38)(36,45)(37,44)(39,47)(40,42)(41,48)(43,46);(2,7,8,9)(3,11,13,12)(17,19)(18,23)(20,29)(21,28)(22,26)(24,32)(25,31)(27,30)(35,43,44,45)(38,47,48,42)

Polycyclic presentation:

2^11|b^2=d,c^2=e,d^2=i,f^2=ij,g^2=jk,h^2=k,b^a=bg,c^a=cf,c^b=cfhk,d^a=dj,d^c=dj,e^a=ei,e^b=eik,f^a=fij,f^c=fj,g^a=gjk,g^b=gk,h^a=hij,h^b=hik

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239321, #72239401
maximal normal subgroup of	#72239321, #72239401

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