#492523

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

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

normal subgroups only

This group has 502 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)(9,10,16,12)(11,21,23,36)(13,27,30,19)(14,26,33,18)(15,28,32,20)(17,29,22,24)(25,39,34,37)(31,38,40,35);(9,11)(10,17)(12,22)(13,29)(14,31)(15,34)(16,23)(18,36)(19,37)(20,38)(21,26)(24,30)(25,32)(27,39)(28,35)(33,40);(2,5)(3,8)(6,7)(9,12,16,10)(11,22,23,17)(13,26,30,18)(14,27,33,19)(15,28,32,20)(21,24,36,29)(25,38,34,35)(31,39,40,37);(9,13)(10,18)(11,23)(12,26)(14,32)(15,33)(16,30)(17,22)(19,28)(20,27)(21,39)(24,40)(29,31)(36,37)

Polycyclic presentation:

2^9|a^2=i,c^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, 399719);