#106623

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

826 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:

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

Polycyclic presentation:

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

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, 13819);