C₂×#7542 #153564

order	512
nilpotency class	6
isoclinism	isoclinic class of #7540
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7542

Subgroups

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂³	SD₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

178 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 130
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 130 points:

(1,2);(3,4,7,13,10,17,34,16)(5,18,20,64,23,68,31,67)(6,24,27,80,30,84,92,83)(8,35,32,100,38,50,102,103)(9,39,33,15,41,56,78,62)(11,42,43,19,47,70,46,73)(12,48,51,94,54,113,53,28)(14,57,55,26,60,89,59,37)(21,75,74,120,76,116,98,126)(22,77,40,66,79,115,99,124)(25,58,85,123,71,63,90,122)(29,95,91,82,97,128,127,118)(36,65,105,117,69,49,101,114)(44,96,107,104,110,130,109,87)(45,72,108,106,111,125,61,86)(52,88,119,129,121,93,81,112);(3,5,10,23)(4,11,17,47)(6,25,30,71)(7,31,34,20)(8,36,38,69)(9,40,41,99)(12,49,54,65)(13,46,16,43)(14,58,60,63)(15,61,62,108)(18,24,68,84)(19,28,73,94)(21,26,76,37)(22,78,79,33)(27,90,92,85)(29,96,97,130)(32,101,102,105)(35,44,50,110)(39,45,56,111)(42,48,70,113)(51,117,53,114)(52,120,121,126)(55,122,59,123)(57,98,89,74)(64,83,67,80)(66,95,124,128)(72,112,125,129)(75,81,116,119)(77,82,115,118)(86,88,106,93)(87,127,104,91)(100,109,103,107)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60760);