C₂×#7391 #150073

order	512
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7391

Subgroups

	subgroup	G/H
center, socle	C₂²	#1333
commutator	C₁₆	C₂²×C₈
Frattini, cosocle	#346	C₂³
prime radical	C₂³	#361
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 246 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);(3,4,7,13,9,15,31,43,10,17,8,32,30,16,34,14)(5,18,20,42,21,51,28,50,22,53,55,65,35,52,33,64)(6,23,25,47,26,48,62,46,27,60,29,12,41,44,54,45)(11,24,36,19,38,56,49,58,40,61,66,59,39,57,37,63);(3,5,10,22)(4,11,17,40)(6,24,27,61)(7,28,8,33)(9,35,30,21)(12,42,47,65)(13,37,32,49)(14,36,43,66)(15,39,16,38)(18,48,53,44)(19,54,59,62)(20,34,55,31)(23,52,60,51)(25,58,29,63)(26,57,41,56)(45,64,46,50)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#150074, C₂×#151105, C₂×#150148, …
maximal normal subgroup of	C₂×#150074, C₂×#151105, C₂×#150148, …
maximal quotient of	C₂×#94059, #72322461, #72322468

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57269);