C₂×#7347 #149547

order	512
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7347

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂²	#1316
prime radical	C₂×#585	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

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

normal subgroups only

This group has 998 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);(3,4,7,13,9,14,28,46,10,15,29,47,8,30,33,45)(5,16,18,48,19,49,53,65,20,40,54,62,31,61,26,58)(6,21,23,55,24,12,39,41,25,42,27,59,32,43,57,66)(11,34,35,17,37,50,44,52,38,60,63,22,36,56,64,51);(3,5)(4,11)(6,22)(7,26)(8,31)(9,19)(10,20)(12,40)(13,44)(14,37)(15,38)(16,43)(17,25)(18,28)(21,49)(23,34)(24,51)(27,60)(29,53)(30,36)(32,52)(33,54)(35,45)(39,50)(41,65)(42,61)(46,63)(47,64)(48,55)(56,57)(58,66)(59,62)

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, 56743);