#72234335

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆² · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²×C₈	C₄²
commutator	C₄	C₂×C₁₆²
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle, prime radical	C₂³	C₂×#1311
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 2048

A minimal faithful permutation representation — 3 generators on 56 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=e,c^2=fk,d^2=g,e^2=h,f^2=k,g^2=i,h^2=j,i^2=k,b^a=bf,d^b=dk,e^a=ek

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234414
maximal normal subgroup of	#72234414
maximal quotient of	#72234340

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