#6358796

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,5)(3,8)(6,7)(9,10)(11,15)(12,19)(13,21)(14,17)(16,18)(20,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(2,6)(3,8)(5,7)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(2,7)(5,6)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#135×AGL(1,ℤ/8), #72342596
maximal normal subgroup of	#135×AGL(1,ℤ/8), #72342596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265992);