#507683

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 555 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=g,e^2=i,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^c=fh,h^c=hi,f^d=fi

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, 414879);