#152834

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

217 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 78 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)(5,6,11,18,13,21,40,20)(7,22,26,59,28,61,65,60)(8,29,30,67,32,58,62,46)(9,33,35,43,37,17,54,55)(10,38,39,53,36,16,23,52)(12,41,34,19,42,56,24,57)(14,31,45,64,48,68,47,25)(15,49,50,66,51,63,44,27);(5,7,23,62,42,66,37,64,40,65,36,32,34,49,35,31,13,28,39,30,12,27,9,25,11,26,10,8,24,63,54,68)(6,14,43,51,57,67,16,22,18,45,17,44,41,58,52,59,21,48,55,15,19,46,38,61,20,47,33,50,56,29,53,60);(2,4)(5,8,9,28,34,64,23,63,11,30,35,65,42,68,10,27,13,32,37,7,24,25,39,49,40,62,54,26,12,31,36,66)(6,15,16,47,41,60,55,67,20,44,53,48,57,61,17,29,21,51,38,45,56,59,43,46,18,50,52,14,19,22,33,58)

Polycyclic presentation:

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

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, 60030);