C₂×#13700 #512948

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13700

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

434 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 4 generators on 66 points:

(1,2);(4,11)(5,18)(6,12)(7,9)(8,22)(13,39)(14,33)(15,45)(16,31)(17,34)(19,49)(20,51)(21,26)(23,37)(24,36)(25,38)(27,28)(29,40)(30,32)(35,47)(41,64)(42,57)(43,56)(44,59)(46,58)(50,61)(52,60)(54,63)(55,62)(65,66);(3,4,12,32,8,15,23,31,7,14,36,58,27,44,25,34,10,17,38,59,28,46,24,33,9,16,37,45,22,30,6,11)(5,13,35,55,20,41,52,57,26,43,61,66,53,65,50,56,21,42,60,64,51,62,47,39,18,40,49,63,48,54,19,29);(3,5)(4,13)(6,19)(7,26)(8,20)(9,18)(10,21)(11,29)(12,35)(14,43)(15,41)(16,40)(17,42)(22,48)(23,52)(24,47)(25,50)(27,53)(28,51)(30,54)(31,57)(32,55)(33,39)(34,56)(36,61)(37,49)(38,60)(44,65)(45,63)(46,62)(58,66)(59,64)

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, 420144);