#72264931

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#368117
commutator	C₂×C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	#7623353	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #355236 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=f,c^2=fgj,d^2=h,e^2=i,g^2=j,j^2=k,c^a=cj,c^b=cg,d^a=dh,d^b=dh,d^c=de,e^a=ei,f^c=fk,g^a=gk,g^b=gj,g^c=gk,g^d=gi,h^c=hi,j^b=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265591
maximal normal subgroup of	#72265591

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