#72322707

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²×C₄²
commutator	C₂²×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
socle	C₂³	C₂²×#346
prime radical	C₂²×#41	M₃₂
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^11|a^2=f,d^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^a=ce,c^b=ci,d^a=djk,f^d=fk,g^a=gk

Conjugacy classes & character table

344 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323148
maximal normal subgroup of	#72323148
maximal quotient of	#72322743

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