#128887

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#329
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	C₄×#329
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36083);