#106210

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,g^2=h,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, 13406);