C₈×#504 #479898

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈² · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₈×#504

Subgroups

	subgroup	G/H
center	C₈×C₁₆	C₂²
commutator	C₂	C₂²×C₈²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

752 conjugacy classes of subgroups; only the 686 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 686 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:

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

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₈×#2171, C₂×C₈×#504, C₈×#3407, …
maximal normal subgroup of	C₈×#2171, C₂×C₈×#504, C₈×#3407, …
maximal quotient of	C₂×C₈×#504, C₁₆×#504, #72226969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387094);