C₂×#7447 #150946

order	512
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7447

Subgroups

	subgroup	G/H
center, socle	C₂²	#1361
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 90 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);(3,4,7,14,32,17,34,15,11,18,8,16,10,37,9,35)(5,19,21,43,38,51,22,56,24,52,36,66,23,57,33,50)(6,25,27,47,61,45,63,58,31,59,28,13,30,44,29,46)(12,26,39,55,41,64,48,54,42,60,65,20,49,53,40,62);(3,5)(4,12)(6,26)(7,22)(8,33)(9,36)(10,38)(11,24)(13,43)(14,48)(15,39)(16,40)(17,49)(18,42)(19,45)(20,29)(21,34)(23,32)(25,51)(27,54)(28,62)(30,64)(31,60)(35,65)(37,41)(44,52)(46,50)(47,66)(53,61)(55,63)(56,58)(57,59)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58142);