C₂×#13679 #512872

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

Aliases

direct product

C₂×#13679

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#124	C₂⁴
prime radical	C₂×#3288	C₂
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

1048 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1048 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);(3,4,8,14)(5,15,19,32)(6,12,22,11)(7,13,25,26)(9,20,29,16)(10,30,28,17)(18,27,23,31)(21,34,24,33);(3,5)(4,9)(6,20)(7,18)(8,19)(10,26)(11,32)(12,15)(13,28)(14,29)(16,22)(17,24)(21,30)(23,25)(27,33)(31,34);(3,5)(4,10)(6,17)(7,23)(8,19)(9,26)(11,32)(12,15)(13,29)(14,28)(16,21)(18,25)(20,24)(22,30)(27,34)(31,33)

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, 420068);