C₂×#13566 #512570

order	512
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13566

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1481 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);(3,4,7,13,9,14,22,28)(5,15,18,26,20,32,34,10)(6,11,21,12,8,23,24,29)(16,30,33,27,19,31,17,25);(3,5)(4,10)(6,19)(7,17)(8,16)(9,20)(11,27)(12,15)(13,30)(14,26)(18,24)(21,34)(22,33)(23,25)(28,31)(29,32);(3,6,9,8)(4,11,14,23)(5,16,20,19)(7,21,22,24)(10,25,26,27)(12,28,29,13)(15,30,32,31)(17,18,33,34)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419766);