#72305384

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁴×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁶
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

At least 6294 conjugacy classes of subgroups; only the 3703 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|b^2=g,c^2=j,d^2=g,f^2=gh,i^2=j,c^a=cj,c^b=ci,d^c=dj,f^a=fgh,f^e=fh,i^b=ij,i^c=ij

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#525, #72307841
maximal normal subgroup of	AGL(1,ℤ/8)×#525, #72307841
maximal quotient of	#72305577

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