C₂×#13708 #512959

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

Aliases

direct product

C₂×#13708

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 74
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 74 points:

(1,2);(3,4,6,8)(5,9,10,7)(11,12,14,18)(13,23,24,52)(15,31,28,53)(16,36,29,61)(17,38,30,22)(19,43,40,25)(20,46,41,34)(21,49,42,33)(26,57,55,74)(27,58,39,54)(32,50,59,72)(35,68,60,64)(37,47,62,71)(44,69,70,66)(45,56,51,65)(48,67,63,73);(3,5,6,10)(4,7,8,9)(12,18)(19,40)(20,41)(21,42)(22,38)(23,52)(31,53)(36,61)(44,70)(45,51)(47,71)(48,63)(50,72)(54,58)(57,74)(64,68);(11,13,14,24)(12,19,18,40)(15,32,28,59)(16,37,29,62)(17,39,30,27)(20,47,41,71)(21,50,42,72)(22,51,38,45)(23,53,52,31)(25,46,43,34)(26,49,55,33)(35,69,60,66)(36,44,61,70)(48,74,63,57)(54,64,58,68)(56,67,65,73)

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, 420155);