C₂×#13615 #512686

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13615

Subgroups

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

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

normal subgroups only

This group has 794 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)(21,47)(23,53)(24,55)(25,48)(26,56)(30,44)(32,62)(34,41)(35,40)(36,49)(37,63)(38,52)(39,58)(42,68)(43,65)(45,60)(46,51)(50,67)(54,69)(57,66)(59,61)(64,70);(7,9)(8,14)(10,21)(11,30)(12,34)(13,25)(15,36)(16,41)(17,44)(18,46)(19,38)(20,29)(22,31)(23,27)(24,28)(26,57)(32,64)(33,37)(35,50)(39,53)(40,66)(42,69)(43,67)(45,70)(47,56)(48,62)(49,58)(51,68)(52,65)(54,60)(55,61)(59,63);(7,10,13,28)(8,15,19,18)(9,21,25,24)(11,26,32,59)(12,27,35,60)(14,36,38,46)(16,39,43,42)(17,40,45,33)(20,47,48,55)(22,49,52,51)(23,50,54,34)(29,56,62,61)(30,57,64,63)(31,58,65,68)(37,44,66,70)(41,53,67,69)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419882);