C₂×#13666 #512825

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13666

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 70
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 70 points:

(1,2);(3,4)(5,6)(7,8,13,19)(9,20,25,51)(10,15,28,18)(11,29,32,63)(12,33,35,41)(14,36,39,22)(16,42,45,31)(17,46,48,27)(21,50,24,57)(23,55,56,69)(26,58,61,62)(30,47,65,70)(34,43,53,68)(37,54,49,52)(38,59,66,64)(40,67,44,60);(7,9)(8,14)(10,21)(11,30)(12,34)(13,25)(15,37)(16,43)(17,47)(18,49)(19,39)(20,29)(22,31)(23,27)(24,28)(26,59)(32,65)(33,38)(35,53)(36,42)(40,55)(41,66)(44,69)(45,68)(46,56)(48,70)(50,58)(51,63)(52,60)(54,67)(57,62)(61,64);(4,6)(7,10,13,28)(8,15,19,18)(9,21,25,24)(11,26,32,61)(12,27,35,46)(14,37,39,49)(16,40,45,44)(17,41,48,33)(20,50,51,57)(22,52,36,54)(23,53,56,34)(29,58,63,62)(30,59,65,64)(31,60,42,67)(38,47,66,70)(43,55,68,69)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420021);