#72226363

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#7318
commutator	C₂²×C₂²⋊C₄	C₂×C₄²
Frattini, cosocle	#34981	C₂³
prime radical	C₂×#34981	C₂²
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ C₂×#34981 ◁ G
LECS	G ▷ #34981 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

9922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=defjk,e^2=jk,b^a=be,c^a=cj,c^b=cj,d^b=dg,d^c=dk,e^a=egjk,e^b=egh,e^d=eik,f^a=fh,f^b=fh,f^d=fijk,g^a=gi,g^b=gj,g^e=gk,h^a=hijk,h^b=hijk,h^d=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226365, #72316386, #72316398
maximal normal subgroup of	#72226365, #72316386, #72316398

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