C₂×#13678 #512873

order	512
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13678

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#124	C₂⁴
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1091 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1091 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);(4,9)(6,17)(10,26)(11,25)(12,29)(13,18)(14,31)(15,21)(20,22)(24,27)(28,32)(30,34);(3,4)(5,10)(6,18)(7,12)(8,13)(9,17)(11,23)(14,32)(15,27)(16,28)(19,34)(20,29)(21,30)(22,25)(24,33)(26,31);(3,5,8,16)(4,10,13,28)(6,19,17,33)(7,21,23,15)(9,24,18,34)(11,27,12,30)(14,22,31,20)(25,26,29,32)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420069);