#491106

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 443 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398302);