#72263161

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1898
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂²×D₄²	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,c^2=ef,d^2=ij,e^2=j,c^a=ce,c^b=ci,d^a=dghi,d^b=dij,d^c=dhij,e^a=ei,e^c=eh,e^d=ei,f^a=fhij,f^c=fh,f^d=fij,f^e=fj,g^c=gij,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263165, #72263166, #72263233, …
maximal normal subgroup of	#72263165, #72263166, #72263233, …

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