#135636

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#431
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,9,6,10,12,14,4,7,13,15,11,16,3,8)(17,18,21,27,20,26,29,23,22,25,31,24,30,32,19,28);(2,7)(8,14)(9,15)(10,16)(17,19,20,21,22,29,30,31)(18,23,26,24,25,28,32,27);(1,3,11,13,4,12,6,5)(2,8,16,15,7,14,10,9)(17,19,20,21,22,29,30,31)(18,24,32,23,25,27,26,28)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=dh,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^b=fg,f^c=fg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42832);