SD₁₆×Q₃₂ #492578

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

Aliases

direct product

SD₁₆×Q₃₂

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×D₄	C₂×D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

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, 399774);