#129381

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8302
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36577);