#72309308

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×C₂²⋊C₄
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂⁶	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

8415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^10|a^2=f,c^2=fhj,e^2=fhj,f^2=j,g^2=j,h^2=j,c^a=cj,e^a=efgj,e^b=eij,e^c=efhj,e^d=efh,f^e=fi,g^a=gi,g^e=gi,h^e=hi

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309309
maximal normal subgroup of	#72309309

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