C₈×#350 #128243

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₈×#350

Subgroups

	subgroup	G/H
center	C₈²	D₄
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₂²⋊C₈
prime radical	C₂×#41	C₄²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 214 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 — 3 generators on 40 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#2171, C₂×C₈×#350
maximal normal subgroup of	C₈×#2171, C₂×C₈×#350
maximal quotient of	C₂×C₈×#350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35439);