C₂×#13663 #512824

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13663

Subgroups

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

792 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 154 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);(3,4)(5,6)(8,14)(9,13)(10,20)(12,22)(15,30)(16,29)(17,32)(18,33)(19,25)(21,26)(23,35)(24,34)(27,28)(31,37)(36,38);(7,8)(9,15)(10,21)(11,16)(12,26)(13,27)(14,20)(17,22)(18,32)(19,31)(23,36)(24,37)(25,38)(28,34)(29,35)(30,33);(4,6)(7,9,11,13)(8,15,16,27)(10,18,23,24)(12,19,25,22)(14,28,29,30)(17,26,31,38)(20,34,35,33)(21,32,36,37)

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, 420020);