#72322687

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#2177
commutator	C₂×C₁₆	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	C₂³×D₁₆	C₂×C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^11|c^2=e,d^2=g,f^2=hj,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=ci,d^a=dk,d^c=djk,f^a=fh,f^c=fh,f^d=fk,g^c=gk,h^a=hj,h^c=hj,j^a=jk,j^c=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323148
maximal normal subgroup of	#72323148
maximal quotient of	#72322689

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