C₂×#7378 #149651

order	512
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7378

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂²	#1326
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ 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);(3,4,7,13,9,14,28,44,10,15,29,45,8,30,32,43)(5,16,18,46,20,48,26,59,21,38,53,62,19,47,55,65)(6,22,23,56,24,12,37,39,25,40,27,60,31,41,58,66)(11,33,34,52,35,61,63,54,36,57,64,17,49,50,42,51);(3,5,8,19,10,21,9,20)(4,11,14,35,15,36,30,49)(6,17,24,51,25,52,31,54)(7,26,28,53,29,55,32,18)(12,38,22,48,41,16,40,47)(13,42,43,64,45,63,44,34)(23,57,58,61,27,33,37,50)(39,65,60,46,66,59,56,62)

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, 56847);