C₂×#13618 #512688

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13618

Subgroups

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

774 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 282 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)(9,20)(10,15)(11,29)(12,33)(13,19)(14,22)(16,31)(17,27)(18,28)(21,48)(23,54)(24,50)(25,49)(26,59)(30,45)(32,47)(34,42)(35,41)(36,51)(37,64)(38,57)(39,53)(40,61)(43,70)(44,52)(46,55)(56,68)(58,69)(60,67)(62,63)(65,66);(7,9,13,24)(8,14,19,39)(10,25,28,21)(11,30,32,65)(12,34,35,58)(15,38,18,36)(16,42,44,69)(17,45,46,66)(20,47,50,29)(22,52,53,31)(23,55,56,27)(26,60,62,64)(33,37,41,67)(40,54,43,68)(48,59,49,63)(51,61,57,70);(7,10,13,28)(8,15,19,18)(9,21,24,25)(11,26,32,62)(12,27,35,55)(14,36,39,38)(16,40,44,43)(17,41,46,33)(20,48,50,49)(22,51,53,57)(23,34,56,58)(29,59,47,63)(30,64,65,60)(31,61,52,70)(37,66,67,45)(42,68,69,54)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419884);