C₂×#7438 #150294

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7438

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂²	#1352
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 74 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);(3,4,7,12)(5,15,17,41)(6,11,21,34)(8,14,26,39)(9,13,25,40)(10,28,29,49)(16,30,44,37)(18,42,47,64)(19,32,24,53)(20,31,48,57)(22,36,51,38)(23,35,50,62)(27,33,55,61)(43,59,54,66)(45,58,46,63)(52,60,56,65);(3,5)(4,10)(6,20,22,49)(7,24,25,54)(8,19)(9,18)(11,32,36,59)(12,37,40,63)(13,31)(14,30)(15,35,42,60)(16,23,45,52)(17,26,47,55)(21,44)(27,43)(28,56)(29,39,57,61)(33,58)(34,41)(38,64)(46,51)(48,50)(53,62)(65,66)

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, 57490);