C₂×#7454 #150963

order	512
nilpotency class	5
isoclinism	isoclinic class of #7453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#7454

Subgroups

	subgroup	G/H
center, socle	C₂²	#1365
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

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

normal subgroups only

This group has 194 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);(3,4,7,14,32,16,27,13,8,35,33,45,38,44,30,64,11,17,28,49,9,36,29,46,10,15,34,48,26,51,6,25)(5,18,20,40,59,43,58,52,21,47,60,12,37,39,19,41,24,55,62,50,22,63,57,53,23,54,61,42,56,66,31,65);(3,5)(4,12,17,42)(6,20)(7,31)(8,23)(9,37)(10,21)(11,24)(13,43,46,63)(14,47,49,54)(15,50,35,40)(16,52,36,53)(18,48,55,45)(19,28)(22,38)(25,39,64,66)(26,59)(27,61)(29,60)(30,62)(32,56)(33,57)(34,58)(41,44,65,51)

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, 58159);