#152868

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13176
prime radical	C₂³	#510
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

183 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60064);