C₂×#7377 #149649

order	512
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7377

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂²	#1326
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

410 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 82 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);(3,4,7,13,9,14,29,44,10,15,30,45,8,31,33,43)(5,16,18,46,20,48,52,66,21,49,53,62,19,47,27,59)(6,22,24,55,25,12,38,39,26,40,28,60,32,41,58,65)(11,34,35,17,36,50,42,51,37,61,64,23,56,57,63,54);(3,5,8,19,10,21,9,20)(4,11,14,36,15,37,31,56)(6,23,25,54,26,17,32,51)(7,27,29,18,30,52,33,53)(12,16,22,47,41,49,40,48)(13,42,43,35,45,63,44,64)(24,34,58,57,28,61,38,50)(39,59,60,46,65,66,55,62)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56845);