#135609

order	512
nilpotency class	4
isoclinism	isoclinic class of #135590
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=di,e^2=ghi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=eh,f^c=fg,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42805);