C₂²×#1416 #512685

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1416

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

806 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 512

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419881);