#106218

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=dh,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13414);