C₄×#1418 #151965

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×#1418

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂×#508	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

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

normal subgroups only

This group has 302 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);(6,11,12,27)(7,15)(8,20)(9,22)(13,31,30,35)(14,32,24,29)(16,21,28,26)(17,23)(18,36)(19,25)(33,34);(5,6)(7,16)(8,21)(9,24)(10,12)(11,25)(13,20)(14,22)(15,27)(17,30)(18,31)(19,28)(23,26)(29,34)(32,36)(33,35)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#13658, C₈×#1418, #72223678, …
maximal normal subgroup of	C₄×#13658, C₈×#1418, #72223678, …
maximal quotient of	C₈×#1418, C₂×C₄×#1418, AGL(1,𝔽₅)×#1418

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59161);