C₂×#7522 #153452

order	512
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7522

Subgroups

	subgroup	G/H
center, socle	C₂²	#1402
commutator	C₄⋊Q₈	C₂²×C₄
Frattini, cosocle	#494	C₂³
prime radical	C₂³	AΣL(1,𝔽₁₆)
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#494 ◁ G
LECS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 325 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);(3,4,7,11,8,13,21,31)(5,12,15,32,16,27,22,30)(6,9,18,23,19,25,33,29)(10,26,28,17,24,20,34,14);(3,5,8,16)(4,9,13,25)(6,17,19,14)(7,15,21,22)(10,27,24,12)(11,28,31,34)(18,20,33,26)(23,32,29,30)

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, 60648);