#109285

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

545 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16481);