#138165

order	512
nilpotency class	4
isoclinism	isoclinic class of #138165
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9133
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3038	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45361);