C₂×#13728 #513046

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#13728

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×D₃₂	C₂
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

558 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 170 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);(4,6)(7,8)(9,19)(10,25)(11,15)(12,31)(13,32)(14,21)(16,20)(17,26)(18,30)(22,39)(23,52)(24,53)(27,54)(28,58)(29,59)(33,48)(34,62)(35,63)(36,44)(37,57)(38,64)(40,43)(41,47)(42,51)(45,56)(46,55)(49,60)(50,69)(61,70)(65,67)(66,68);(7,9,20,43,12,23,45,63,30,51,64,70,57,68,27,48,11,22,44,67,29,50,28,49,13,24,46,62,26,47,10,21)(8,14,25,41,17,34,55,53,32,60,58,69,59,65,36,39,15,33,54,66,37,61,38,42,18,35,56,52,31,40,16,19);(3,4,5,6)(8,15)(14,33)(16,36)(17,37)(18,32)(19,39)(25,54)(31,59)(34,61)(35,60)(38,55)(40,65)(41,66)(42,53)(52,69)(56,58)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420242);