C₂×#7521 #153451

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

Aliases

direct product

C₂×#7521

Subgroups

	subgroup	G/H
center, socle	C₂²	#1402
commutator	C₄⋊Q₈	C₂²×C₄
Frattini, cosocle	#494	C₂³
prime radical	C₂×#555	C₄
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

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

normal subgroups only

This group has 341 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);(3,4,6,11,7,13,19,31)(5,14,16,28,15,25,34,29)(8,12,23,32,21,33,20,30)(9,18,24,22,10,26,27,17);(4,9)(5,15)(10,13)(11,28)(12,14)(16,34)(17,22)(18,26)(24,30)(25,33)(27,32)(29,31)

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, 60647);