C₂×#13606 #512644

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13606

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	C₂×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

832 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 186 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);(3,4,6,12)(5,10,15,24)(7,11,17,26)(8,13,18,27)(9,19,22,20)(14,25,28,34)(16,23,30,33)(21,31,32,29);(4,9)(6,17)(7,20)(10,23)(11,13)(15,30)(16,29)(18,19)(21,25)(22,26)(24,34)(28,31);(3,5,8,14)(4,10,13,25)(6,15,18,28)(7,16,20,29)(9,21,11,23)(12,24,27,34)(17,30,19,31)(22,32,26,33)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#512649, C₂×#152139, C₂×#152143
maximal normal subgroup of	C₂×#512649, C₂×#152139, C₂×#152143
maximal quotient of	#72293103, #72293106, #72293200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419840);