#145539

order	512
nilpotency class	4
isoclinism	isoclinic class of #145538
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,7)(3,8)(4,5)(9,10)(11,16)(12,17)(13,30)(14,31)(15,20)(18,27)(19,28)(21,39)(22,35)(23,40)(24,33)(25,37)(26,34)(29,38)(32,36);(1,2)(3,4)(5,7)(6,8)(9,11,15,25)(10,16,20,37)(12,26,29,23)(13,21,27,22)(14,32,28,24)(17,34,38,40)(18,35,30,39)(19,33,31,36);(1,3,4,2,6,7,5,8)(10,17)(11,21)(12,27)(13,29)(14,28)(16,33)(18,19)(20,38)(22,25)(23,32)(24,26)(30,31)(35,39)(36,37)

Polycyclic presentation:

2^9|b^2=i,c^2=d,d^2=g,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fhi,h^b=hi,e^c=egh,f^c=fghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52735);