#512813

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂×#41	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=de,c^b=cei

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420009);