C₂×#13689 #512909

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13689

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×D₃₂	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(4,6)(7,8)(9,13)(10,14)(11,24)(12,25)(15,21)(16,22)(17,20)(18,26)(19,36)(23,32)(27,33)(28,34)(29,37)(30,35)(31,38);(3,4,5,6)(8,13)(14,26)(15,27)(16,28)(17,25)(24,36)(29,31)(30,32);(8,14)(10,21)(11,22)(12,20)(13,26)(15,16)(17,32)(18,33)(19,34)(23,38)(24,29)(25,30)(27,28)(31,36)(35,37)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420105);