C₂×#13572 #512582

order	512
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13572

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
socle	C₂²	C₂×AΣL(1,𝔽₁₆)
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1491 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1491 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);(3,4,7,13,9,15,23,31)(5,16,19,27,20,29,34,10)(6,11,21,28,8,14,25,12)(17,33,18,32,24,30,22,26);(3,5)(4,10)(6,17)(7,22)(8,24)(9,20)(11,26)(12,29)(13,30)(14,32)(15,27)(16,28)(18,23)(19,21)(25,34)(31,33);(3,6,9,8)(4,11,15,14)(5,17,20,24)(7,21,23,25)(10,26,27,32)(12,13,28,31)(16,33,29,30)(18,34,22,19)

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, 419778);