#507707

order	512
nilpotency class	4
isoclinism	isoclinic class of #501527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,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, 414903);