#137816

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9086
prime radical	C₂⁴	#131
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,f^d=fi

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, 45012);