C₂×#7381 #150050

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7381

Subgroups

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	D₄⋊C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 106 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);(3,4,7,13,9,15,32,44,8,16,31,45,6,25,28,56,10,17,33,46,29,12,39,41,27,14,47,43,30,42,58,64)(5,18,20,49,22,38,53,62,23,40,54,63,26,48,57,65,24,50,55,66,19,11,34,35,21,37,52,61,51,36,59,60);(3,5)(4,11)(6,26)(7,20)(8,23)(9,22)(10,24)(12,40)(13,35)(14,48)(15,37)(16,36)(17,38)(18,25)(19,29)(21,27)(28,57)(30,51)(31,54)(32,53)(33,55)(34,39)(41,63)(42,50)(43,65)(44,61)(45,60)(46,62)(47,52)(49,56)(58,59)(64,66)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57246);