#72258353

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#116351
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
prime radical	#62811	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=efi,c^2=g,d^2=j,b^a=be,c^a=cd,c^b=cfhi,d^a=dh,d^b=dhi,d^c=dh,e^a=ei,e^b=ei,e^c=ehij,e^d=ej,f^b=fi,f^c=fhj,g^a=ghj,g^b=gh,g^e=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258361, #72258400, #72258445
maximal normal subgroup of	#72258361, #72258400, #72258445

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