C₂×#7526 #153488

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7526

Subgroups

	subgroup	G/H
center, socle	C₂²	#1404
commutator	#125	C₂²×C₄
Frattini, cosocle	#491	C₂³
prime radical	C₂³	AΣL(1,𝔽₁₆)
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#491 ◁ G
LECS	G ▷ #491 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 34 points:

(1,2);(3,4,7,12,8,14,5,15)(6,18,20,28,22,29,34,10)(9,13,17,32,24,33,23,31)(11,19,27,25,30,16,26,21);(3,5,8,7)(4,10,14,28)(6,19,22,16)(9,23,24,17)(11,15,30,12)(13,27,33,26)(18,31,29,32)(20,21,34,25)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60684);