#72239031

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1438
commutator	C₂³×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
socle	C₂²	#7555
prime radical	C₂×#3444	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=d,c^2=ehj,d^2=ij,g^2=j,h^2=j,b^a=bf,c^a=ce,c^b=cdfg,d^a=dj,d^c=di,e^b=ei,f^b=fj,f^c=fij,g^b=gj,g^c=gj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239036, #72239043, #72239170, …
maximal normal subgroup of	#72239036, #72239043, #72239170, …
maximal quotient of	#72257652

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