C₄×#1392 #146350

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₄×#1392

Subgroups

	subgroup	G/H
center	C₄²	#99
commutator	C₂³	C₄³
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	C₂×#342
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 246 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);(5,6,9,14,11,16,10,15)(7,17,19,32,21,34,20,33)(8,22,23,31,24,13,25,29)(12,26,27,18,28,35,30,36);(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,30)(8,18,23,35,24,36,25,26)(13,17,29,33,22,34,31,32)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53546);