C₄×#1419 #151966

order	512
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#1419

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#13658, C₈×#1419, #72223677, …
maximal normal subgroup of	C₄×#13658, C₈×#1419, #72223677, …
maximal quotient of	C₈×#1419, C₂×C₄×#1419, AGL(1,𝔽₅)×#1419

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59162);