C₂×#7449 #150949

order	512
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7449

Subgroups

	subgroup	G/H
center, socle	C₂²	#1361
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 90 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);(3,4,7,13,30,16,32,42,10,17,31,15,9,33,8,14)(5,18,20,49,34,51,58,66,23,52,57,41,22,59,21,50)(6,24,26,12,40,43,53,45,29,61,64,46,28,44,27,60)(11,25,35,19,38,54,47,65,39,63,37,56,48,62,36,55);(3,5,10,23)(4,11,17,39)(6,25,29,63)(7,21,31,58)(8,20,32,57)(9,34,30,22)(12,41,46,49)(13,36,15,47)(14,35,42,37)(16,48,33,38)(18,43,52,44)(19,53,56,27)(24,59,61,51)(26,55,64,65)(28,54,40,62)(45,66,60,50)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58145);