#139697

order	512
nilpotency class	4
isoclinism	isoclinic class of #139572
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1910
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
socle	C₂	#9492
prime radical	#2010	C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46893);