#131434

order	512
nilpotency class	4
isoclinism	isoclinic class of #131432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8404
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60937	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

641 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38630);