#512150

order	512
nilpotency class	4
isoclinism	isoclinic class of #511779
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1051 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,d^b=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419346);