C₂×#7423 #150253

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7423

Subgroups

	subgroup	G/H
center, socle	C₂²	#1347
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(1,2);(3,4)(5,16,18,47)(6,22,25,46)(7,29)(8,32)(9,35)(10,15)(11,17)(12,26,41,27)(13,24)(14,31)(19,52,20,53)(21,54,36,48)(23,43)(28,61,59,55)(30,64,33,40)(34,44)(37,62)(38,56)(39,51)(42,60,65,58)(45,50)(49,66)(57,63);(3,5,10,18)(4,11,15,39)(6,23,25,49)(7,30,8,33)(9,36,31,21)(12,40,41,64)(13,43,44,66)(14,45,35,38)(16,46,47,22)(17,26,51,27)(19,24,20,34)(28,62,59,57)(29,63,32,37)(42,53,65,52)(48,55,54,61)(50,60,56,58)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57449);