#72235124

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁶
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

At least 6600 conjugacy classes of subgroups; only the 5276 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 24 points:

(11,17)(12,20)(15,23)(16,22)(18,21)(19,24);(1,2)(3,5)(4,6)(7,8)(15,19)(23,24);(15,19)(23,24);(1,3)(2,5)(4,7)(6,8)(10,14)(11,18)(16,22)(17,21);(2,6)(5,8)(9,10,13,14)(11,19,18,15)(12,16,20,22)(17,24,21,23);(2,6)(3,7)(9,11)(10,15)(12,21)(13,18)(14,19)(16,24)(17,20)(22,23)

Polycyclic presentation:

2^11|e^2=k,j^2=k,e^b=eg,e^c=ei,e^d=ek,f^a=fj,f^b=fgh,f^c=fhi,f^d=fgik,f^e=fk,g^f=gk,h^e=hk,h^f=hk,i^f=ik,j^a=jk,j^f=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236860, #72237656, #72251984, …
maximal normal subgroup of	#72236860, #72237656, #72251984, …
automorphism group of	#1773, #1946, #1975, …

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