Q₁₆×Q₃₂ #492778

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

Aliases

direct product

Q₁₆×Q₃₂

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 392 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16);(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34);(17,33,25,41)(18,34,26,42)(19,35,27,43)(20,36,28,44)(21,37,29,45)(22,38,30,46)(23,39,31,47)(24,40,32,48)

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, 399974);