#145400

order	512
nilpotency class	4
isoclinism	isoclinic class of #145348
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
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 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,21)(12,17)(13,16)(14,19)(15,23)(18,28)(20,27)(22,30)(24,38)(25,34)(26,32)(29,39)(31,37)(33,40)(35,36);(1,2,6,8)(3,5,7,4)(9,11)(10,15)(12,23)(13,24)(14,22)(16,31)(17,21)(18,29)(19,34)(20,30)(25,27)(26,37)(28,36)(32,38)(33,35)(39,40);(1,3,4,2,6,7,5,8)(10,16)(11,22)(13,28)(14,27)(15,29)(17,32)(18,20)(19,33)(21,35)(23,25)(24,37)(26,40)(30,38)(31,34)

Polycyclic presentation:

2^9|b^2=g,c^2=di,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,e^c=egh,f^c=fi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52596);