#152881

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60077);