C₂×#13655 #512771

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13655

Subgroups

	subgroup	G/H
center	C₂×C₈	D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₄×D₁₆
prime radical	C₂×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 178 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);(3,4,8,14,10,16,31,15)(5,17,20,47,22,48,21,53)(6,12,24,39,25,40,9,32)(7,26,28,56,30,58,62,57)(11,34,36,52,38,51,45,19)(13,23,42,54,44,55,43,29)(18,46,49,65,50,66,33,41)(27,35,59,63,61,64,60,37);(3,5)(4,11)(6,18)(7,27)(8,20)(9,33)(10,22)(12,35)(13,41)(14,36)(15,45)(16,38)(17,26)(19,29)(21,31)(23,34)(24,49)(25,50)(28,59)(30,61)(32,37)(39,63)(40,64)(42,46)(43,66)(44,65)(47,56)(48,58)(51,55)(52,54)(53,57)(60,62);(3,6,8,24,10,25,31,9)(4,12,14,39,16,40,15,32)(5,18,20,49,22,50,21,33)(7,23,28,54,30,55,62,29)(11,35,36,63,38,64,45,37)(13,26,42,56,44,58,43,57)(17,46,47,65,48,66,53,41)(19,27,34,59,52,61,51,60)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#512773
maximal normal subgroup of	C₂×#512773
maximal quotient of	C₂×C₈×D₃₂, C₂×C₈×SD₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419967);