#512789

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

347 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 248 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231957, #72231958, #72231959, …
maximal normal subgroup of	#72231957, #72231958, #72231959, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419985);