#72287139

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₈²	C₂⁵
socle	C₂²	C₂²×#1580
prime radical	#7623990	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ #479949 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

3284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 64 points:

(2,8)(9,13)(10,22)(11,23)(12,24)(25,28)(26,31)(27,32)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,57)(46,58)(47,54)(48,52)(49,55)(50,53)(51,56)(59,63)(60,62)(61,64);(1,2,7,8)(3,13,17,9)(4,10,15,22)(5,11,21,23)(6,12,14,24)(16,28,19,25)(18,26,30,31)(20,27,29,32)(33,35,39,38)(34,40,43,44)(36,45,49,48)(37,46,51,50)(41,52,55,57)(42,53,56,58)(47,60,61,59)(54,63,64,62);(1,3,5,15,6,16,18,29,7,17,21,4,14,19,30,20)(2,9,11,10,12,25,26,27,8,13,23,22,24,28,31,32)(35,38)(36,47,49,61)(37,50)(40,44)(41,54,55,64)(42,53)(45,59,48,60)(46,51)(52,62,57,63)(56,58);(1,4,18,3,14,29,5,19,7,15,30,17,6,20,21,16)(2,10,26,13,24,32,11,25,8,22,31,9,12,27,23,28)(36,48,49,45)(37,51)(41,52,55,57)(42,56)(46,50)(47,60,61,59)(53,58)(54,62,64,63);(33,36,37,47,38,48,50,59,39,49,51,61,35,45,46,60)(34,41,42,54,40,52,53,63,43,55,56,64,44,57,58,62)

Polycyclic presentation:

2^11|b^2=jk,c^2=f,d^2=fh,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,b^a=bjk,c^b=cjk,e^c=eg,e^d=ei,f^e=fk,g^c=gi,g^d=gk,i^c=ik

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287443, #72322318, #72323169
maximal normal subgroup of	#72287443, #72322318, #72323169

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