#133272

order	512
nilpotency class	4
isoclinism	isoclinic class of #133271
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	#3444	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

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

normal subgroups only

This group has 522 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,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,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40468);