#72308543

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#354203
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	C₂×#28143	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×D₄×#97 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^11|c^2=f,d^2=h,e^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bg,d^a=de,d^c=djk,e^a=ei,e^b=eijk,e^c=ek,f^d=fk,g^d=gijk,h^a=hik,h^c=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308663
maximal normal subgroup of	#72308663

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