#151963

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₄×D₁₆
prime radical	C₂×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 288 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,18,14,21,13,39)(7,22,25,55,29,56,28,65)(8,16,31,20,12,38,33,52)(9,19,34,49,35,51,32,46)(11,17,36,48,30,47,37,50)(15,40,42,61,45,59,66,24)(23,53,58,43,27,64,60,67)(26,54,62,68,57,41,63,44);(2,4)(5,7,11,26,12,27,34,40,10,25,36,62,33,60,35,61,14,29,30,57,8,23,32,59,13,28,37,63,31,58,9,24)(6,15,19,43,20,44,50,65,39,66,46,53,16,41,47,56,21,45,51,67,52,68,48,55,18,42,49,64,38,54,17,22);(5,8,10,31,14,12,13,33)(6,16,18,20,21,38,39,52)(7,23,25,58,29,27,28,60)(9,30,34,37,35,11,32,36)(15,41,42,44,45,54,66,68)(17,46,48,19,47,49,50,51)(22,53,55,43,56,64,65,67)(24,57,40,63,61,26,59,62)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=i,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,h^a=hi

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, 59159);