#72340148

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#13561
commutator	#127	C₂³×C₄
Frattini, cosocle	#492	C₂⁴
socle	C₂	C₂×#13561
prime radical	#6348073	C₂
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ #60639 ◁ G
LECS	G ▷ #492 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2053 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=e,e^2=hi,g^2=i,h^2=j,b^a=bf,c^b=cij,d^a=di,d^b=dij,e^a=eg,e^c=ej,e^d=ej,f^a=fj,f^b=fgi,f^d=fj,f^e=fhij,g^a=gi,g^b=gh,g^f=gi,h^a=hi,h^b=hj,h^f=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340151, #72340152, #72340154, …
maximal normal subgroup of	#72340151, #72340152, #72340154, …

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