#72226613

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×#120
commutator, Frattini, cosocle	C₂⁶	C₂⁵
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂⁶ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂ ▷ C₁

2103164 conjugacy classes of subgroups; only the 1696 normal subgroups are shown.

normal subgroups only

This group has 2103164 conjugacy classes of subgroups — too many to draw. Only its 1696 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(29,32);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(27,31)(29,32);(3,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,23)(22,24);(1,2,4,5)(3,8,7,6)(10,14)(13,16)(19,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,22)(19,23)(25,27)(26,29)(28,31)(30,32)

Polycyclic presentation:

2^11|d^2=hi,e^2=hi,d^a=dg,d^b=dh,d^c=dj,e^a=ef,e^b=ejk,e^c=ehj,e^d=ehi,f^d=fk,g^e=gk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226615
maximal normal subgroup of	#72226615
maximal quotient of	#72267568, #72267617

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