#151909

order	512
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	#532	C₂³
socle	C₂	C₄×AΣL(1,𝔽₁₆)
prime radical	#3018	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

455 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=i,g^2=i,b^a=bd,f^a=fgi,g^a=gi,h^a=hi,d^b=dg,g^b=gh,h^b=hi,f^d=fhi,g^d=gi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59105);