C₂×#7365 #149607

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7365

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂²	#1322
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

258 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 98 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);(3,4,6,12,7,13,20,33,8,14,21,34,22,35,44,56,9,11,19,29,23,30,42,51,24,31,43,52,45,53,61,64)(5,10,16,25,17,27,39,46,18,28,40,47,41,50,60,62,15,26,36,32,37,48,57,54,38,49,58,55,59,63,66,65);(3,5)(4,10,11,26)(6,16)(7,17)(8,18)(9,15)(12,32,29,25)(13,27,30,48)(14,28,31,49)(19,36)(20,39)(21,40)(22,41)(23,37)(24,38)(33,54,51,46)(34,55,52,47)(35,50,53,63)(42,57)(43,58)(44,60)(45,59)(56,65,64,62)(61,66)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56803);