C₂×#13682 #512891

order	512
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13682

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂⁴
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1045 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 114 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);(4,10)(5,14)(6,17)(7,20)(12,30)(13,18)(15,34)(16,22)(19,24)(21,32)(25,27)(26,33);(3,4,9,13)(5,11,16,29)(6,18,17,10)(7,21,20,30)(8,12,23,32)(14,33,22,26)(15,28,24,31)(19,27,34,25);(3,5)(4,11)(6,19)(7,22)(8,24)(9,16)(10,25)(12,31)(13,29)(14,20)(15,23)(17,34)(18,27)(21,26)(28,32)(30,33)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420087);