C₂×#13742 #513313

order	512
nilpotency class	6
isoclinism	isoclinic class of #13742
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13742

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator, Frattini, cosocle	C₃₂	C₂⁴
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

302 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#153598, D₃×#13742, #72286831, …
maximal normal subgroup of	C₂×#153598, #72286831, #72286862, …
maximal quotient of	D₃×#13742, #72286862

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420509);