#145104

order	512
nilpotency class	4
isoclinism	isoclinic class of #145015
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 337 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=egi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52300);