#128344

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

436 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(9,10,14,19,15,12,17,26)(11,20,23,38,24,34,30,40)(13,25,28,39,29,18,33,35)(16,31,32,22,21,37,36,27);(1,2,3,5,4,6,7,8)(9,11)(10,16)(12,21)(13,27,29,22)(14,30)(15,24)(17,23)(18,34,25,20)(19,36)(26,32)(28,37,33,31)(35,38,39,40);(9,12,14,26,15,10,17,19)(11,21,30,32,24,16,23,36)(13,25,28,39,29,18,33,35)(20,37,40,22,34,31,38,27)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,e^2=i,f^2=h,b^a=bd,d^a=di,f^a=fg,d^b=dg,e^b=ei,g^b=gi,f^d=fi

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, 35540);