#133263

order	512
nilpotency class	4
isoclinism	isoclinic class of #133263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8522
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#60830	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 615 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=h,e^2=i,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=fhi,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, 40459);