#72287239

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄²
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₄×D₄²
prime radical	#6358421	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=g,c^2=f,d^2=fi,e^2=h,f^2=i,g^2=ij,h^2=j,b^a=bg,c^b=cgij,d^a=dij,d^c=di,e^a=eij,e^c=eh,e^d=ej,g^a=gij,g^c=gij,h^c=hj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287387, #72319360, #72320885, …
maximal normal subgroup of	#72287387, #72319360, #72320885, …

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