C₂×#13560 #512561

order	512
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#13560

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
socle	C₂²	C₂×AΣL(1,𝔽₁₆)
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1499 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1499 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419757);