#492525

order	512
nilpotency class	4
isoclinism	isoclinic class of D₈×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 126 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)(4,7)(6,8)(10,15)(11,19)(12,24)(13,27)(16,20)(17,36)(18,32)(22,23)(25,29)(26,38)(28,30)(31,37)(34,40)(35,39);(9,10,14,18)(11,20,23,39)(12,25,28,36)(13,29,27,17)(15,30,32,24)(16,33,35,21)(19,31,22,40)(26,37,38,34);(2,5)(3,8)(6,7);(9,11,14,23)(10,16,18,35)(12,26,28,38)(13,19,27,22)(15,31,32,40)(17,37,29,34)(20,36,39,25)(21,30,33,24)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399721);