#142909

order	512
nilpotency class	4
isoclinism	isoclinic class of #142404
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,12,18)(7,19,22,36)(8,24,9,27)(10,17,11,16)(13,30,32,21)(14,25,15,26)(20,33,28,35)(23,31,29,34);(5,7,11,23,12,22,10,29)(6,13,17,33,18,32,16,35)(8,20,25,21,9,28,26,30)(14,31,27,19,15,34,24,36);(2,4)(7,20)(8,25)(9,26)(10,11)(13,31)(14,27)(15,24)(16,17)(19,35)(21,29)(22,28)(23,30)(32,34)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50105);