C₄×#1426 #151997

order	512
nilpotency class	5
isoclinism	isoclinic class of #1426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄×#1426

Subgroups

	subgroup	G/H
center	C₂×C₄	#367
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#367
prime radical	C₂³	C₂×SD₃₂
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

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

normal subgroups only

This group has 208 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);(6,11,13,24)(7,16)(8,20)(9,21)(12,27)(14,28,31,36)(15,34,32,26)(17,22)(18,30)(19,29)(23,35,33,25);(5,6,12,28,8,14,7,11,21,32,16,25,17,23,27,34,10,13,30,36,22,31,19,24,9,15,29,35,20,33,18,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#13658, C₈×#1426, #72223674, …
maximal normal subgroup of	C₄×#13658, C₈×#1426, #72223674, …
maximal quotient of	C₈×#1426, C₂×C₄×#1426, AGL(1,𝔽₅)×#1426

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59193);