#151916

order	512
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	#532	C₂³
socle	C₂	C₄×AΣL(1,𝔽₁₆)
prime radical	#3018	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59112);