#133273

order	512
nilpotency class	4
isoclinism	isoclinic class of #133273
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8524
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#3508	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40469);